Carleson Measure Estimates and the Dirichlet Problem for Degenerate Elliptic Equations Hofmann, Steve; Le, Phi; Morris, Andrew

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Carleson measure estimates and the Dirichlet problem for degenerate elliptic equations Hofmann, Steve; Le, Phi; Morris, Andrew

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Download date: 01. Feb. 2019 CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM FOR DEGENERATE ELLIPTIC EQUATIONS

STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

Abstract. We prove that the Dirichlet problem for degenerate elliptic equations n+1 div(A∇u) = 0 in the upper half-space (x, t) ∈ R+ is solvable when n ≥ 2 and p n the boundary data is in Lµ(R ) for some p < ∞. The coefficient matrix A is only assumed to be measurable, real-valued and t-independent with a degenerate bound and ellipticity controlled by an A2-weight µ. It is not required to be symmetric. The result is achieved by proving a Carleson measure estimate for all bounded solutions in order to deduce that the degenerate elliptic measure is n in A∞ with respect to the µ-weighted Lebesgue measure on R . The Carleson measure estimate allows us to avoid applying the method of ϵ-approximability, which simplifies the proof obtained recently in the case of uniformly elliptic coefficients. The results have natural extensions to Lipschitz domains.

Contents

1. Introduction 1 2. Preliminaries 5 3. Estimates for Maximal Operators 11 4. The Carleson Measure Estimate 16 5. Solvability of the Dirichlet Problem 32 5.1. Boundary estimates for solutions 33 5.2. Deﬁnition and properties of degenerate elliptic measure 33 5.3. Preliminary estimates for degenerate elliptic measure 35 5.4. The A∞-estimate for degenerate elliptic measure 38 5.5. The square function and non-tangential maximal function estimates 43 References 47

1. Introduction

We consider the Dirichlet boundary value problem for the degenerate elliptic n+1 equation div(A∇u) = 0 in the upper half-space R+ when n ≥ 2 and which we

Date: December 3, 2018. 2010 Mathematics Subject Classification. 35J25, 35J70, 42B20, 42B25. Key words and phrases. Square functions, non-tangential maximal functions, harmonic measure, Radon–Nikodym derivative, Carleson measure, divergence form elliptic equations, Dirichlet problem, A2 Muckenhoupt weights, reverse Holder¨ inequality. 1 2 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS make precise below. The boundary Rn × {0} is identified with Rn and we adopt the n+1 n notation X = (x, t) for points X ∈ R+ with coordinates x ∈ R and t ∈ (0, ∞). The gradient ∇ := (∇x, ∂t) and divergence div := divx +∂t are with respect to all (n + 1)- coordinates. The coefficient A denotes an (n + 1) × (n + 1)-matrix of measurable, n+1 real-valued and t-independent functions on R+ . The matrix A(x):= A(x, t) is not required to be symmetric. We suppose that there exist constants 0 < λ ≤ Λ < ∞ n and an A2-weight µ on R such that the degenerate bound and ellipticity (1.1) |⟨A(x)ξ, ζ⟩| ≤ Λµ(x)|ξ||ζ| and ⟨A(x)ξ, ξ⟩ ≥ λµ(x)|ξ|2 hold for all ξ, ζ ∈ Rn+1 and almost every x ∈ Rn. We use ⟨·, ·⟩ and | · | to denote the n Euclidean inner-product and norm. An A2-weight µ on R refers to a non-negative locally integrable function µ : Rn → [0, ∞] such that ( ˆ )( ˆ ) 1 1 1 µ n = µ < ∞, [ ]A2(R ) : sup (x) dx dx Q |Q| Q |Q| Q µ(x) Rn | | where supQ denotes the supremum over all´ cubes Q in with volume Q . We µ µ = µ also use to denote the measure (Q): ´ Q (x) dx and consider the Lebesgue p n p 1/p space Lµ(R ) with the norm ∥ f ∥ p n := ( | f | dµ) for all p ∈ [1, ∞). There ffl Lµ(R ) ´ Rn ffl ´ µ = µ −1 µ = | |−1 is also the notation Q f d : (Q) Q f d whilst Q f : Q Q f (x) dx. If µ is identically 1, then A is called uniformly elliptic. The solvability of the Dirichlet problem for general non-symmetric coefficients in that case was obtained only recently by Hofmann, Kenig, Mayboroda and Pipher in [HKMP]. The result in dimension n = 1 had been obtained previously by Kenig, Koch, Pipher and Toro in [KKoPT]. These results assert that for each uniformly elliptic coefficient matrix A, there exists some p < ∞ for which the Dirichlet problem is solvable for Lp-boundary data. Conversely, counterexamples in [KKoPT] show that for each p < ∞, there exists a uniformly elliptic coefficient matrix A for which the Dirichlet problem is not solvable for Lp-boundary data. In contrast, solvability of the Dirichlet problem for symmetric coefficients in the uniformly elliptic case is well-understood, and we mention only that it was obtained by Jerison and Kenig in [JK] for Lp-boundary data when 2 ≤ p < ∞. The solvability of the Dirichlet problem in the uniformly elliptic case has also been established for a variety of complex coefficient structures (see, for instance, [AS, HKMP, HMM]). A significant portion of that theory was recently extended to the degenerate elliptic case by Auscher, Rosen´ and Rule in [ARR] for L2- boundary data. That extension did not include, however, the results for general non-symmetric coefficients in [HKMP]. This paper complements the progress made in [ARR] by extending the solvability obtained for the Dirichlet problem in [HKMP] to the degenerate elliptic case. n+1 n For solvability on the upper half-space R+ , the A2-weight µ on R is extended n+1 µ , = µ R µ n+1 = µ Rn to the t-independent A2-weight (x t): (x) on (and [ ]A2(R ) [ ]A2( )). We then say that u is a solution of the´ equation div(A∇u) = 0 in an open set n+1 1,2 Ω ⊆ R u ∈ W Ω n+1 ⟨A∇u, ∇Φ⟩ = when µ,loc( ) and R+ 0 for all smooth com- Φ ∈ ∞ Ω µ pactly supported functions Cc ( ). The solution space is the local -weighted 1,2 Sobolev space Wµ,loc defined in Section 2. The convergence of solutions to boundary data is afforded by estimates for the non-tangential maximal function N∗u of CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 3 solutions u, defined by n (N∗u)(x):= sup |u(y, t)| ∀x ∈ R , (y,t)∈Γ(x) n+1 where the cone Γ(x):= {(y, t) ∈ R+ : |y − x| < t}. If p ∈ (1, ∞), then the Dirichlet p n problem for Lµ(R )-boundary data, or simply (D)p,µ, is said to be solvable when p for each f ∈ Lµ(Rn), there exists a solution u such that  n+1 div(A∇u) = 0 in R+ , p n (D)p,µ N∗u ∈ Lµ(R ),  limt→0 u(·, t) = f, p where the limit is required to converge in Lµ(Rn)-norm and in the non-tangential n sense whereby limΓ(x)∋(y,t)→(x,0) u(y, t) = f (x) for almost every x ∈ R . Note that this definition of solvability is distinct from well-posedness, which requires that such solutions are unique. We are able to obtain a uniqueness result for solutions that converge uniformly to 0 at infinity, but the question of well-posedness more generally remains open (see Theorem 5.34 and the preceding discussion). n A non-negative Borel measure ω on a cube Q0 in R is said to be in the A∞-class with respect to µ, written ω ∈ A∞(µ), when there exist constants C, θ > 0, which we call the A∞(Q0)-constants, such that ( )θ µ(E) ω(E) ≤ C ω(Q) µ(Q) for all cubes Q ⊆ Q0 and all Borel sets E ⊆ Q. This is a scale-invariant version of the absolute continuity of ω with respect to µ. It is well-known, at least in the uniformly elliptic case, that solvability of the Dirichlet problem for Lp-boundary data for some p < ∞ is equivalent to the property that an adapted harmonic measure (elliptic measure) belongs to A∞ with respect to the Lebesgue measure on Rn (see Theorem 1.7.3 in [K]). In the degenerate case, an adapted harmonic measure ωX, n+1 which we call degenerate elliptic measure, can also be defined at each X ∈ R+ (see Section 5). We prove that this degenerate elliptic measure is in A∞ with respect to µ and then deduce the solvability of (D)p,µ stated in the theorem below. This n requires the notation associated with cubes Q in R whereby xQ and ℓ(Q) denote the centre and side length of Q, respectively, and XQ := (xQ, ℓ(Q)) denotes the n+1 corkscrew point in R+ relative to Q. Theorem 1.2. If n ≥ 2 and the t-independent coefficient matrix A satisfies the degenerate bound and ellipticity in (1.1) for some constants 0 < λ ≤ Λ < ∞ and n an A2-weight µ on R , then there exists p ∈ (1, ∞) such that (D)p,µ is solvable. Moreover, on each cube Q in Rn, the degenerate elliptic measure ω := ωXQ ⌊Q ω ∈ µ λ Λ µ satisfies A∞( ) with A∞(Q)-constants that depend only on n, , and [ ]A2 . In contrast to the proof of solvability in the uniformly elliptic case in [HKMP], we avoid the need to apply the method of ϵ-approximability by first establishing the Carleson measure estimate in the theorem below. This crucial estimate facilitates the main results of the paper. The connection between the Carleson measure estimate and solvability was first established in the uniformly elliptic case by Kenig, 4 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

Kirchheim, Pipher and Toro in [KKiPT], and we follow their approach here, adapting it to the degenerate elliptic setting (see Lemma 5.24 below). In particular, the A∞-property of degenerate elliptic measure is obtained by combining the Carleson measure estimate (1.4) with the notion of good ϵ-coverings introduced in [KKoPT]. Theorem 1.3. If n ≥ 2 and the t-independent coefficient matrix A satisfies the degenerate bound and ellipticity in (1.1) for some constants 0 < λ ≤ Λ < ∞ and n ∞ n+1 n+1 an A2-weight µ on R , then any solution u ∈ L (R+ ) of div(A∇u) = 0 in R+ satisfies the Carleson measure estimate ˆ ˆ ℓ(Q) 1 2 dt 2 (1.4) sup |t∇u(x, t)| dµ(x) ≤ C∥u∥∞, Q µ(Q) 0 Q t λ Λ µ where C depends only on n, , and [ ]A2 . Using the Carleson measure estimate in this way allows us to bypass the need to establish norm-equivalences between the non-tangential maximal function N∗u and the square function S u of solutions u, defined by (¨ ) / dµ(y) dt 1 2 (S u)(x):= |t∇u(y, t)|2 ∀x ∈ Rn, Γ(x) µ(∆(x, t)) t where the surface ball ∆(x, t):= {y ∈ Rn : |y − x| < t}. It was shown by Dahlberg, Jerison and Kenig in [DJK], however, that such estimates are a consequence of the A∞-property of degenerate elliptic measure, which provides the following result. Theorem 1.5. If n ≥ 2 and the t-independent coefficient matrix A satisfies the degenerate bound and ellipticity in (1.1) for some constants 0 < λ ≤ Λ < ∞ and n n+1 an A2-weight µ on R , then any solution of div(A∇u) = 0 in R+ satisfies

∥ ∥ p ≤ ∥ ∥ p ∀ ∈ , ∞ , S u Lµ(Rn) C N∗u Lµ(Rn) p (0 ) n+1 and if, in addition, u(X0) = 0 for some X0 ∈ R+ , then

∥ ∥ p ≤ ∥ ∥ p ∀ ∈ , ∞ , N∗u Lµ(Rn) C S u Lµ(Rn) p (0 ) λ Λ µ where C depends only on X0, p, n, , and [ ]A2 . The paper is structured as follows. Technical preliminaries concerning weights and degenerate elliptic operators are in Section 2 whilst estimates for weighted maximal operators are in Section 3. The Carleson measure estimate in Theorem 1.3 is obtained in Section 4. The degenerate elliptic measure is constructed in Section 5 and then the A∞-estimates in Theorem 1.2 are deduced as part of Theorem 5.30. The square function and non-tangential maximal function estimates in Theorem 1.5 are included in the more general result in Theorem 5.31 whilst the solvability of the Dirichlet problem in Theorem 1.2 is finally deduced in Theorem 5.34, where a uniqueness result is also obtained. We state and prove our results in the upper half-space, but we note that they extend immediately to the case that the domain is the region above a Lipschitz graph, by a well-known pull-back technique which preserves the t-independence of the coefficients. In turn, our results concerning the A∞-property of degenerate elliptic measure may then be extended to the case of a bounded star-like Lipschitz CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 5 domain, with radially independent coefficients, by a standard localization argument using the maximum principle. The convention is adopted whereby C denotes a finite positive constant that may change from one line to the next. For a, b ∈ R, the notation a ≲ b means that a ≤ Cb whilst a ≂ b means that a ≲ b ≲ a. We write a ≲p b when a ≤ Cb and we wish to emphasize that C depends on a specified parameter p.

2. Preliminaries

We dispense with some technical preliminaries concerning general Ap-weights µ for p ∈ (1, ∞) and degenerate elliptic operators on Rn for n ∈ N. All cubes Q and balls B in Rn are assumed to be open (except in Section 5.4 where the standard dyadic cubes S in D(Rn) are assumed to be closed to provide genuine coverings of Rn). For α > 0, let αQ and αB denote the concentric dilates of Q and B respectively. n n For x ∈ R and r > 0, define the ball B(x, r):= {y ∈ R : |y − x| < r}. An Ap-weight refers to a non-negative locally integrable function µ on Rn with the property that ( ffl )( ffl ) − −1/(p−1) p 1 µ Rn = µ µ < ∞ [ ]Ap( ) : supQ Q Q . The measure associated with such a weight satisfies the doubling property µ α ≤ µ αnpµ (2.1) ( B) [ ]Ap (B) for all α ≥ 1 (see, for instance, Section 1.5 in Chapter V of [S2]). 1,p For an open set Ω ⊆ Rn, the Sobolev space Wµ (Ω) is defined as the completion, p ∞ in the ambient space Lµ(Ω), of the normed space of all f ∈ C (Ω) with finite norm ˆ ˆ ∥ ∥p = | |p µ + |∇ |p µ < ∞. (2.2) f 1,p : f d f d Wµ (Ω) Ω Ω 1,p p The embedding of the completion Wµ (Ω) in Lµ(Ω) relies on the Ap-property of µ µ−1/(p−1) 1 Ω the weight (to the extent that it implies both and are in Lloc( )), which 1,p ∞ ensures that if ( f j) j is a Wµ (Ω)-Cauchy sequence in C (Ω) converging to 0 in p 1,p Lµ(Ω), then ( f j) j converges to 0 in Wµ (Ω)-norm (see Section 2.1 in [FKS]). ∞ 1,p Therefore, since C (Ω) is dense in Wµ (Ω), the gradient extends to a bounded 1,p p 1,p operator ∇ : Wµ (Ω) → Lµ(Ω, Rn), thereby extending (2.2) to all f ∈ Wµ (Ω). 1,p Ω ∞ Ω 1,p Ω The Sobolev space W0,µ ( ) is defined as the closure of Cc ( ) in Wµ ( ). It can 1,p Rn = 1,p Rn be shown that W0,µ ( ) Wµ ( ) by following the proof in the unweighted case from Proposition 1 of Chapter V in [S1] but instead using Lemma 2.2 in [ARR] to p Rn 1,p Ω deduce the convergence of the regularization in Lµ( ). The local space Wµ,loc( ) ∈ p Ω ∈ 1,p Ω′ is then defined as the set of all f Lµ,loc( ) such that f Wµ ( ) for all open sets Ω′ with compact closure Ω′ ⊂ Ω (henceforth denoted Ω′ ⊂⊂ Ω). Finally, the weighted Sobolev and Poincare´ inequalities obtained for continuous functions in Theorems 1.2 and 1.5 in [FKS] have the following immediate extensions. Theorem 2.3. Let n ≥ 2 and suppose that B ⊂ Rn denotes a ball with radius r(B). n If p ∈ (1, ∞) and µ is an Ap-weight on R , then there exists δ > 0 such that ( ) / n +δ ( ) / 1 (p( n−1 )) 1 p p( n +δ) p (2.4) | f | n−1 dµ ≲ r(B) |∇ f | dµ B B 6 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

∈ 1,p for all f W0,µ (B), and ( ) ( ) 1/p 1/p p p (2.5) | f (x) − cB| dµ ≲ r(B) |∇ f | dµ B B {ﬄ ﬄ } ∈ 1,p ∈ µ, for all f Wµ (B) and cB B f d B f , where the implicit constants depend µ only on n, p and [ ]Ap . The estimates also hold when the ball B and the radius r(B) are replaced by a cube Q and the sidelength ℓ(Q).

n For n ∈ N, constants 0 < λ ≤ Λ < ∞ and an A2-weight µ on R , let E(n, λ, Λ, µ) denote the set of all n × n-matrices A of measurable real-valued functions on Rn satisfying the degenerate bound and ellipticity (2.6) |⟨A(x)ξ, ζ⟩| ≤ Λµ(x)|ξ||ζ| and ⟨A(x)ξ, ξ⟩ ≥ λµ(x)|ξ|2 for all ξ, ζ ∈ Rn and almost every x ∈ Rn. These properties allow us to define 2 2 2 Lµ,Ω : Dom(Lµ,Ω) ⊆ Lµ(Ω) → Lµ(Ω) as the maximal accretive operator in Lµ(Ω) associated with the bilinear form defined by ˆ ˆ 1 (2.7) aΩ( f, g):= ⟨A∇ f, ∇g⟩ = ⟨ µ A∇ f, ∇g⟩ dµ Ω Ω , ∈ 1,2 Ω L 2 Ω for all f g W0,µ( ). The domain of µ,Ω is dense in Lµ( ), and in particular 1,2 Dom(Lµ,Ω) = { f ∈ W (Ω) : sup ∈ ∞ Ω |aΩ( f, g)|/∥g∥ 2 Ω < ∞}, 0,µ g Cc ( ) Lµ( ) with ˆ (2.8) (Lµ,Ω f )g dµ = aΩ( f, g) Ω ∈ L ∈ 1,2 Ω L for all f Dom( µ,Ω) and g W0,µ( ). It is equivalent to define µ,Ω as the com- 1 ∗ position − divµ,Ω( µ A∇) of unbounded operators, where − divµ,Ω is the adjoint ∇ ∇ 1,2 Ω ⊆ 2 Ω → 2 Ω, Rn of the closed densely-defined operator : W0,µ( ) Lµ( ) Lµ( ), that is ˆ ˆ (2.9) (− divµ,Ω f)g dµ = ⟨f, ∇g⟩ dµ Ω Ω ∈ = ∇∗ ∈ 1,2 Ω for all f Dom(divµ,Ω): Dom( ) and g W0,µ( ). In view of (2.7) and (2.8), 1 1 we have the formal identities divµ,Ω = µ divΩ µ and Lµ,Ω = − µ divΩ(A∇). Now let Ω = Q for some cube Q ⊂ Rn and denote the space of bounded linear 1,2 −1,2 1,2 ⊆ 2 ⊆ −1,2 functionals on W0,µ(Q) by W0,µ (Q). The inclusions W0,µ(Q) Lµ(Q) W0,µ (Q) ∈ 2 ℓ are interpreted in the´ standard way by identifying f Lµ(Q) with the functional f ℓ = µ ∈ 1,2 defined by f (g): Q f g d for all g W0,µ(Q). Thus, setting ˆ

Lµ,Q f (g):= aQ( f, g) and − divµ,Q f(g):= ⟨f, ∇g⟩ dµ Q , ∈ 1,2 ∈ 2 , Rn L for all f g W0,µ(Q) and f L (Q ), we obtain an extension of µ,Q from (2.8) 1,2 −1,2 to a bounded invertible operator from W0,µ(Q) onto W0,µ (Q), and an extension of CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 7

2 −1,2 divµ,Q from (2.9) to a bounded operator from Lµ(Q) into W0,µ (Q). The surjectivity of Lµ,Q relies on (2.4) and the Lax–Milgram Theorem. These deﬁnitions imply that

−1 ∥∇L ∥ n ≲ ∥ ∥ n µ,Q divµ,Q f Lµ2(Q,R ) f Lµ2(Q,R )

∈ 2 , Rn 1,2 for all f Lµ(Q ). The topological direct sum or W0,µ(Q)-Hodge decomposition 2 , Rn = { 1 A∇ ∈ 1,2 } ⊕ { ∈ 2 , Rn = } (2.10) Lµ(Q ) µ g : g W0,µ(Q) h Lµ(Q ) : divµ,Q h 0 = − 1 A∇L−1 + + 1 A∇L−1 = 1 A∇ + follows by writing f µ µ,Q divµ,Q f (f µ µ,Q divµ,Q f) : µ g h, = − L L−1 = since then divµ,Q h divµ,Q f µ,Q µ,Q divµ,Q f 0. This decomposition also p extends to Lµ(Q, Rn) for all p ∈ [2, 2 + ϵ) and some ϵ > 0 by recent work of Le in [L], although we do not need it here. n 1 Now let Ω = R and consider divµ := divµ,Rn as in (2.9) so Lµ := − divµ( µ A∇) 1/2 2 n is maximal accretive, thus having a maximal accretive square root Lµ , in Lµ(R ). The solution of the Kato square root problem in [AHLMT] was recently extended to degenerate elliptic equations by Cruz-Uribe and Rios in [CR3]. This shows that 1/2 1,2 n 1/2 1,2 n ∥L ∥ n ≂ ∥∇ ∥ n n ∈ R L = R µ f Lµ2(R ) f Lµ2(R ,R ) for all f Wµ ( ), hence Dom( µ ) Wµ ( ). 2 n The operator Lµ is also injective and type-S ω+ in Lµ(R ) for some ω ∈ (0, π/2), ∞ o 2 n so it has a bounded H (S θ+)-functional calculus in Lµ(R ) for each θ ∈ (ω, π), o where S θ+ := {z ∈ C \{0} : | arg z| < θ}. See Section 2.2 of [A] for the uniformly elliptic case and Theorems F and G in [ADM] for the general theory. An equivalent property is the validity of the quadratic estimate ˆ ∞ 2 dt 2 2 n (2.11) ∥ψ(tLµ) f ∥ n ≂ ∥ f ∥ n ∀ f ∈ L (R ) Lµ2(R ) Lµ2(R ) µ 0 t o α −β for each holomorphic ψ on S θ+ satisfying |ψ(z)| ≲ min{|z| , |z| } for some α, β > 0, 2 n where the bounded operator ψ(tLµ) on Lµ(R ) is defined by a Cauchy integral. More generally, the relationship between bounded holomorphic functional calculi and quadratic estimates is developed in the seminal articles [Mc] and [CDMY]. 2 n The functional calculus then defines a bounded operator φ(Lµ) on Lµ(R ) for o φ ∥φ L ∥ n n ≲ ∥φ∥ each bounded holomorphic function on S θ+ and ( µ) Lµ2(R )→Lµ2(R ) θ ∞. Another consequence is that −Lµ generates a holomorphic contraction semigroup −ζLµ 2 n −tLµ −tLµ −tLµ (e )ζ∈ o ∪{ } on L (R ), thus e f ∈ Dom(Lµ) and ∂ (e f ) = Lµe f S π/2−ω 0 µ t 2 n for all f ∈ Lµ(R ) and t > 0. The functional calculus also extends to define 2 n o an unbounded operator ϕ(Lµ) on Lµ(R ) for each holomorphic function ϕ on S θ+ satisfying |ϕ(z)| ≲ max{|z|α, |z|−β} for some α, β > 0, but the algebra homomorphism property of the functional calculus (ϕ1(Lµ)ϕ2(Lµ) = (ϕ1ϕ2)(Lµ)) must then be interpreted in the sense of unbounded linear operators. This allows us to interpret both the semigroup and the square root of Lµ in terms of the functional calculus in order to justify some otherwise formal manipulations, beginning with (2.15) in the proof of the following corollary of the solution of the Kato problem in [CR3].

Theorem 2.12. Let n ≥ 1 and suppose that A ∈ E(n, λ, Λ, µ) for some constants n 1 0 < λ ≤ Λ < ∞ and an A2-weight µ on R . The operator Lµ := − divµ( µ A∇) 8 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS satisﬁes ˆ ∞ −t2Lµ 2 dt 2 (2.13) ∥tLµe f ∥ n ≂ ∥∇ f ∥ n n Lµ2(R ) Lµ2(R ,R ) 0 t and ˆ ∞ 2 −t2Lµ 2 dt 2 (2.14) ∥t ∇ , Lµe f ∥ n n+ ≲ ∥∇ f ∥ n n x t Lµ2(R ,R 1) Lµ2(R ,R ) 0 t ∈ 1,2 Rn λ Λ µ for all f Wµ ( ), where the implicit constants depend only on n, , and [ ]A2 .

Proof. The functional calculus of Lµ justiﬁes the identity

−t2Lµ 1/2 −t2Lµ 1/2 −(t2/2)Lµ −(t2/2)Lµ (2.15) Lµe f = Lµ e Lµ f = e Lµe f

1/2 for all f ∈ Dom(Lµ ) and t > 0. The ﬁrst equality in (2.15), the quadratic estimate in (2.11) and the solution of the Kato problem in [CR3] imply that ˆ ˆ ∞ ∞ τ −t2Lµ 2 dt 1/2 −τLµ 1/2 2 d ∥tLµe f ∥ n = ∥(τLµ) e L f ∥ n Lµ2(R ) µ Lµ2(R ) 0 t 0 τ 1/2 2 ≂ ∥L f ∥ n µ Lµ2(R ) 2 ≂ ∥∇ f ∥ n n Lµ2(R ,R ) 1/2 1,2 n for all f ∈ Dom(Lµ ) = Wµ (R ), which proves (2.13). ∞ o The bounded H (S θ+)-functional calculus of Lµ implies the uniform estimate

∥ ∇ −t2Lµ ∥2 = ∥ ∂ −t2Lµ ∥2 + ∥ ∇ −t2Lµ ∥2 t x,te g 2 Rn,Rn+1 t te g 2 Rn t xe 2 Rn,Rn Lµ( ) Lµ( ) ˆ Lµ( ) 2 −t2Lµ 2 2 −t2Lµ −t2Lµ ≲ ∥t Lµe g∥ n + t ⟨A∇ e g, ∇ e g⟩ Lµ2(R ) x x Rn 2 2 −t2Lµ −t2Lµ ≲ ∥g∥ n + ∥t Lµe g∥ 2 Rn ∥e g∥ 2 Rn Lµ2(R ) Lµ( ) Lµ( ) 2 ≲ ∥g∥ n Lµ2(R ) 2 n for all g ∈ Lµ(R ) and t > 0. Thus, the second equality in (2.15) and the vertical square function estimate in (2.13), which we have already proved, imply that ˆ ∞ ˆ ∞ 2 −t2Lµ 2 dt −(t2/2)Lµ 2 dt 2 ∥t ∇ , Lµe f ∥ n n+ ≲ ∥tLµe f ∥ n ≲ ∥∇ f ∥ n n x t Lµ2(R ,R 1) Lµ2(R ) Lµ2(R ,R ) 0 t 0 t 1,2 n for all f ∈ Wµ (R ), which proves (2.14). □

Now let us return to the case when Ω ⊆ Rn is an arbitrary open set and suppose n 1 ∞ that f : Ω → R is a measurable function for which µ f ∈ L (Ω). A solution of n the inhomogeneous equation´ div(A∇u) = div f in Ω ⊆ R refers to any function ∈ 1,2 Ω ⟨A∇ − , ∇Φ⟩ = Φ ∈ ∞ Ω u Wµ,loc( ) such that Rn u f 0 for all Cc ( ). All solutions u of the homogeneous equation div(A∇u) = 0 in Ω are locally bounded and Holder¨ continuous in the sense that ( ) 1/2 2 (2.16) ∥u∥L∞(B) ≲ |u| dµ 2B CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 9 and there exists α > 0 such that ( )α ( ) / |x − y| 1 2 (2.17) |u(x) − u(y)| ≲ |u|2 dµ ∀x, y ∈ B, r(B) 2B and if, in addition, u ≥ 0 almost everywhere on Ω, there is the Harnack inequality (2.18) sup u ≲ inf u, B B for all balls B of radius r(B) such that 2B ⊆ Ω, where α and the implicit constants λ Λ µ depend only on n, , and [ ]A2 . These properties follow from Corollary 2.3.4, Lemma 2.3.5 and Theorem 2.3.12 in [FKS] by observing that the proofs do not use the assumption therein that A is symmetric. The estimates also hold when the balls B are replaced by (open) cubes Q, and also when the dilate 2B is replaced by C0B for any C0 > 1, provided the implicit constants are understood to depend on C0. The following local boundedness estimate for solutions of the inhomogeneous equation is needed in Lemma 4.3, although only for p = 2. This is a simpler version of Theorem 8.17 in [GT], which we have adapted to degenerate elliptic equations. In fact, the result for p ≥ 2 is already proven in [FKS] by combining Corollary 2.3.4 with estimates (2.3.7) and (2.3.13) therein. The proof is included here for the readers convenience and since it implies (2.16) as a specical case, which in turn is the well-known starting point for establishing (2.17). Theorem 2.19. Let n ≥ 2 and suppose that A ∈ E(n, λ, Λ, µ) for some constants n n 0 < λ ≤ Λ < ∞ and an A2-weight µ on R . Let Ω ⊆ R denote an open set n 1 ∞ and suppose that f : Ω → R is a measurable function such that µ f ∈ L (Ω). If p ∈ (1, ∞) and div(A∇u) = div f in Ω, then ( ) 1/p p 1 (2.20) ∥u∥L∞(B) ≲ |u| dµ + r(B)∥ µ f∥L∞(Ω) 2B for all balls B of radius r(B) > 0 such that 2B ⊆ Ω, where the implicit constant λ Λ µ depends only on p, n, , and [ ]A2 . Proof. Suppose that div(A∇u) = div f in Ω and consider a ball B such that 2B ⊆ Ω. ∞ 1 First, assume that u is non-negative and in L (2B). Let ϵ > 0, set k = r(B)∥ µ f∥L∞(Ω) andu ¯ϵ := u + k + ϵ. Let Br denote the ball concentric to B with radius r > 0 and recall the index δ > 0 from the Sobolev inequality in Theorem 2.3. We claim that if γ ∈ [p, ∞) and r(B) ≤ r1 < r2 ≤ 2r(B), then ( ) / γ n +δ ( ) ( ) /γ 1 ( ( n−1 )) 2/γ 1 γ n +δ ( n−1 ) r1 γ (2.21) u¯ϵ dµ ≲ γ u¯ϵ dµ , r2 − r1 Br1 Br2 λ Λ µ where the implicit constant depends only on p, n, , and [ ]A2 . To prove (2.21), η ∈ ∞ Ω η Ω → , η ≡ η ≡ Ω \ ﬁx Cc ( ) such that : [0 1], 1 on Br1 , 0 on Br2 and ∥∇η∥ ≤ / − β = γ − ν = η2 β ν ∈ 1,2 Ω ∞ 2 (r2 r1). Set : 1 and : u¯ϵ . Note that W0,µ( ) with β 2 β−1 ∇v = 2η∇ηu¯ϵ + βη u¯ϵ ∇u, since 0 < ϵ ≤ u¯ϵ(x) ≤ ∥u∥ ∞ + k + ϵ < ∞ for almost every x ∈ 2B, thus ˆ L (2B) ˆ β 2 β−1 ⟨A∇u − f, 2η∇ηu¯ϵ ⟩ = − ⟨A∇u − f, βη u¯ϵ ∇u⟩. Rn Rn 10 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

We then use this identity and Cauchy’s inequality with σ > 0 to obtain ˆ ˆ 2 β−1 2 2 β−1 η u¯ϵ |∇u| dµ ≲λ η u¯ϵ ⟨A∇u, ∇u⟩ Rn Rn ˆ ˆ −1 β 2 β−1 = −2β ηu¯ϵ ⟨A∇u − f, ∇η⟩ + η u¯ϵ ⟨f, ∇u⟩ Rnˆ Rn ˆ −1 β 1 2 β−1 1 ≲Λ (p−1) ηu¯ϵ (|∇u| + | µ f|)|∇η| dµ + η u¯ϵ | µ f||∇u| dµ ˆ Rn ˆ Rn 2 β−1 2 −1 β+1 2 ≲p σ η u¯ϵ |∇u| dµ + σ u¯ϵ |∇η| dµ Rˆn ˆ Rn β+1 2 2 β+1 + u¯ϵ |∇η| dµ + (η/r(B)) u¯ϵ dµ Rˆn Rn ˆ 2 β−1 2 −1 2 β+1 + σ η u¯ϵ |∇u| dµ + σ (η/r(B)) u¯ϵ dµ, Rn Rn where in the second inequality we used the assumption that β := γ − 1 ≥ p − 1 and 1 in the final inequality we used the fact that | µ f| ≤ k/r(B) ≤ u¯ϵ/r(B) on Ω. Next, choose σ > 0 small enough, depending only on p, λ and Λ, to deduce that ˆ ˆ ˆ β−1 2 β+1 2 2 1 β+1 |∇ | µ ≲ ,λ,Λ |∇η| + η/ µ ≲ µ, u¯ϵ u d p u¯ϵ ( ( r(B)) ) d 2 u¯ϵ d Rn (r2 − r1) Br1 Br2 where in the final inequality we used the fact that r(B) ≥ r2 − r1. Now combine this estimate with the Sobolev inequality (2.4) and recall that β := γ − 1 to obtain ( ) / n +δ 1 ( n−1 ) γ( n +δ) n−1 µ ≲ 2 |∇ (β+1)/2 |2 µ u¯ϵ d r1 (¯uϵ ) d B B r1 r1 2 β−1 2 ≲ ((β + 1)r1) u¯ϵ |∇u| dµ B ( ) r1 2 r1 γ ≲ γ u¯ϵ dµ, r2 − r1 Br2 λ Λ µ where the implicit constants depend only on p, n, , and [ ]A2 , proving (2.21). We now apply the Moser iteration technique to prove (2.20). Set χ := n + δ ( ) n−1 ffl /q q 1 and define Φ(q, r):= u¯ϵ dµ for q, r > 0. Estimate (2.21) implies that Br ( ) 2/γ r1 Φ(γχ, r1) ≤ Cγ Φ(γ, r2) r2 − r1 λ Λ µ where C depends only on p, n, , and [ ]A2 , and it follows by induction that ∑ − ∑ − − 2 m 1 χ−k 2 m 1 kχ−k Φ(pχm, (1 + 2 m)r(B)) ≤ (4Cp) p k=0 (2χ) p k=0 Φ(p, 2r(B)) ≲ Φ(p, 2r(B)) for all m ∈ N. This shows that ( ) 1/p m p ∥u¯ϵ∥ ∞ = lim Φ(pχ , r(B)) ≲ Φ(p, 2r(B)) = u¯ dµ L (B) →∞ ϵ m 2B CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 11 and therefore ( ) ( ) 1/p 1/p p p 1 ∥u∥L∞(B) ≤ ∥u¯ϵ∥L∞(B) ≲ u¯ϵ dµ ≲ u dµ + r(B)∥ µ f∥L∞(Ω) + ϵ 2B 2B for all ϵ > 0, which implies (2.20). Finally, it remains to remove the assumption that u is non-negative and bounded. This is achieved by settingu ¯ϵ := max{u, 0} + k + ϵ andu ¯ϵ := − min{u, 0} + k + ϵ respectively and in each case adjusting the proof above to incorporate the truncated test function ν := η2h (¯uϵ)¯uϵ, where N { xβ−1, x ≤ N + k + ϵ, hN(x):= (N + k + ϵ)β−1, x > N + k + ϵ. We leave the standard details to the reader. □

The following self-improvement property for Carleson measures will be used in conjunction with the local Holder¨ continuity estimate for solutions in (2.17). The result is proved in the unweighted case in Lemma 2.14 in [AHLT]. In that proof, the Lebesgue measure on Rn can in fact be replaced by any doubling measure, since the Whitney decomposition of open sets can be adapted to any such measure (see, for instance, Lemma 2 in Chapter I of [S2]). The result below then follows. n Lemma 2.22. Let n ≥ 1 and suppose that µ is an A2-weight on R . Let α, β0 > 0 and suppose that (vt)t>0 is a collection of H¨oldercontinuous functions on a cube Q ⊂ Rn satisfying ( )α |x − y| 0 ≤ v (x) ≤ β and |v (x) − v (y)| ≤ β t 0 t t 0 t for all x, y ∈ Q. If there exists η ∈ (0, 1], β > 0 and, for each cube Q′ ⊆ Q, a measurable set F′ ⊂ Q′ such that ˆ ˆ l(Q′) µ ′ ≥ ηµ ′ 1 µ dt ≤ β, (F ) (Q ) and ′ vt(x) d (x) µ(Q ) 0 F′ t then ˆ ˆ 1 ℓ(Q) dt vt(x) dµ(x) ≲α,η β + β0, µ(Q) 0 Q t α η µ where the implicit constant depends only on , , n and [ ]A2 .

3. Estimates for Maximal Operators η eη We obtain estimates for a variety of maximal operators (Mµ, D∗,µ, N∗ and N∗,µ) 1 adapted to an A2-weight µ and degenerate elliptic operators Lµ := − divµ( µ A∇) on Rn for n ≥ 2. These will be used to prove the Carleson measure estimate from Theorem 1.3 in Section 4. We ﬁrst deﬁne the maximal operators Mµ and D∗,µ by

∈ 1 Rn ∈ 1,2 Rn ∈ Rn for all f Lµ,loc( ), g Wµ,loc( ) and x . The usual unweighted and centred Hardy–Littlewood maximal operator is abbreviated by M. The maximal operator p Mµ is bounded on Lµ(Rn) for all p ∈ (1, ∞) and satisﬁes the weak-type estimate ( ) n −1 µ { ∈ R | | > κ} ≲ κ ∥ ∥ n ∀κ > (3.1) x : Mµ f (x) f Lµ1(R ) 0 1 n for all f ∈ Lµ(R ) (see, for instance, Theorem 1 in Chapter I of [S2]). There is also the following weak-type estimate for the maximal operator D∗,µ. Lemma 3.2. Let n ≥ 2. If µ is an A -weight on Rn, then ( 2 ) n −2 2 (3.3) µ {x ∈ R : |D∗,µ f (x)| > κ} ≲ κ ∥∇ f ∥ n n ∀κ > 0 Lµ2(R ,R ) ∈ 1,2 Rn µ for all f Wµ ( ), where the implicit constant depends only on n and [ ]A2 . ∈ ∞ Rn Proof. If f Cc ( ), then a version of Morrey’s inequality (see, for instance, Theorem 3.5.2 in [Mo]) shows that | f (x) − f (y)| ≲ M(∇ f )(x) + M(∇ f )(y) |x − y| for almost every x, y ∈ Rn, hence ( ) / 2 1 2 D∗,µ f (x) ≲ M(∇ f )(x) + Mµ[M(∇ f )] (x) .

Estimate (3.3) then follows from the weak-type bound for Mµ in (3.1), the fact that 2 n M is bounded on Lµ(R ) (see, for instance, Theorem 1 in Chapter V of [S2]) and ∞ Rn 1,2 Rn □ the density of Cc ( ) in Wµ ( ). η eη We now define the non-tangential maximal operators N∗ and N∗,µ, for η > 0, by ( ) 1/2 η eη 2 N∗ u(x):= sup |u(y, t)|, N∗,µv(x):= sup |v(z, t)| dµ(z) (y,t)∈Γη(x) (y,t)∈Γη(x) B(y,ηt) , Rn+1 ·, ∈ 2 Rn > for all measurable functions u v on + (such that v( t) Lµ,loc( ) for a.e. t 0) n n+1 and x ∈ R , where Γη(x):= {(y, t) ∈ R+ : |y− x| < ηt} is the conical non-tangential n+1 approach region in R+ with vertex at x and aperture η. Now suppose that A ∈ E(n, λ, Λ, µ), as defined by (2.6). In particular, since A has real-valued coefficients, there exists an integral kernel W (x, y) such that ˆ t −tLµ (3.4) e f (x) = Wt(x, y) f (y) dµ(y), Rn 2 n for all f ∈ Lµ(R ), and there exists constants C1, C2 > 0 such that ( ) 2 C1 |x − y| (3.5) |Wt(x, y)| ≤ √ exp −C2 µ(B(x, t)) t > , ∈ Rn ∈ ∞ Rn for all t 0 and x y . This was proved by Cruz-Uribe and Rios for f Cc ( ) under the assumption that A is symmetric (see Theorem 1 and Remark 3 in [CR2]). The symmetry assumption can be removed, however, by following their proof and applying the Harnack inequality for degenerate parabolic equations obtained by Ishige in Theorem A of [I], which does not require symmetric coefficients, instead of the version recorded in Proposition 3.8 of [CR1]. The results also extend to 2 n f ∈ Lµ(R ) by density, Schur’s Lemma and the doubling property of µ. CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 13

1 We now consider the semigroup generated by Lµ := − divµ( µ A∇) with elliptic 2 −t2Lµ homogeneity (t replaced by t ) and denoted by Pt := e in the estimates below. Lemma 3.6. Let n ≥ 2 and suppose that A ∈ E(n, λ, Λ, µ) for some constants n 0 < λ ≤ Λ < ∞ and an A2-weight µ on R . Let p ∈ (1, ∞) and suppose that µ is n n also an Ap-weight on R . If x ∈ R , η > 0 and α ≥ 1, then −1 2 p 2/p (3.7) sup |(ηt) [Pηt( f − cB(x,αηt))](y)| ≲α [Mµ(|∇ f | )(x)] (y,t)∈Γη(x) {ffl ffl } ∈ 1,p Rn ∈ µ, for all f Wµ ( ) and cB(x,αηt) B(x,αηt) f d B(x,αηt) f , and η 2 p 2/p (3.8) |N∗ (∂tPt f )(x)| ≲η [Mµ(|∇ f | )(x)] , −1 η 2 p 2/p (3.9) |η N∗ (∂ Pη f )(x)| ≲ [Mµ(|∇ f | )(x)] , t t ( ) eη 2 p 2/p 2 (3.10) |N∗,µ(∇xPηt f )(x)| ≲ Mµ [Mµ(|∇ f | )] (x) + Mµ(|∇ f | )(x), ∈ 1,2 Rn ∩ 1,p Rn for all f Wµ ( ) Wµ,loc( ), where the implicit constants depend only on n, λ Λ µ µ α η , , p, [ ]A2 and [ ]Ap , as well as on in (3.7) and on in (3.8). ffl ∈ Rn , ∈ Γη ∈ 1,2 Rn ∩ 1,p Rn = Proof. Letffl x ,(y t) (x), f Wµ ( ) Wµ,loc( ), fB(x,t) : B(x,t) f and ˜ = µ ffi η = α ≥ fB(x,t) : B(x,t) f d . To prove (3.7), it su ces to assume that 1 and 1. We j j−1 set C0(t):= B(x, αt) and define the dyadic annulus C j(t):= B(x, 2 αt)\B(x, 2 αt) for all j ∈ N. The Gaussian kernel estimates in (3.4) and (3.5) imply that ˆ

To estimate I0, note that B(x, αt) ⊆ B(y, (1 + α)t) and apply theﬄ doubling property µ p = of , followed by the Lµ-Poincare´ inequality in (2.5) with cB B(x,αt) f , to obtain ( ) 1/p −1 p p 1/p I0 ≲α t | f (z) − fB(x,αt)| dµ(z) ≲ |∇ f | dµ ≲ [Mµ(|∇ f | )(x)] . B(x,αt) B(x,αt)

To estimate I j, for each j ∈ N, expand f (z) − fB(x,αt) as a telescoping sum to write ( j − j−1α− 2 µ(B(x, 2 αt)) − ≤ C2(2 1) 1 | − ˜ | µ I j C1e t f fB(x,2 jαt) d µ(B(y, t)) B(x,2 jαt) ∑j ) + | ˜ − ˜ | + | ˜ − | fB(x,2iαt) fB(x,2i−1αt) fB(x,αt) fB(x,αt) i=1

j ∑j − j−1α− 2 µ(B(y, (1 + 2 α)t)) − ≲ C2(2 1) 1 | − ˜ | µ e t f fB(x,2iαt) d µ(B(y, t)) , iα i=0 B(x 2 t) ( ) ∑j 1/p − j−1α− 2 ≲ e C2(2 1) (1 + 2 jα)2n 2iα |∇ f |p dµ , iα i=0 B(x 2 t) 14 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

−C4 j n j p 1/p ≲α e 4 [Mµ(|∇ f | )(x)] , where the second inequality relies on the inclusion B(x, 2 jαt) ⊆ B(y, (1 + 2 jα)t), whilst the third inequality uses the doubling propertyffl of µ in (2.1) with p = 2, and p = µ the Lµ-Poincare´ inequality in (2.5) with cB B(x,2iαt) f d . Altogether, we have ( ∑∞ ) −1 −C4 j n j p 1/p p 1/p |t [Pt( f − fB(x,αt))](y)| ≲α e 4 [Mµ(|∇ f | )(x)] ≲ [Mµ(|∇ f | )(x)] , j= 0 ffl ffl = = µ which proves (3.7) when cB(x,αt) B(x,αt) f . The proof when cB(x,αt) B(x,αt) f d follows as above by replacing fB(x,αt) with f˜B(x,αt), since (2.5) can still be applied. To prove (3.8) and (3.9), suppose that η > 0. The Gaussian kernel estimate for −tLµ e e in (3.5) implies that t∂ P f (y) has an integral kernel W 2 (y, z) satisfying t t ( )t | − |2 e C1 y z |W 2 (y, z)| ≤ exp −C t µ(B(y, t)) 2 t2 ´ e , µ = ∈ Rn > and the conservation property Rn Wt2 (y z) d (y) 0 for all z and t 0. This follows from Theorem 5 in [CR2], where the assumption that A is symmetric can be removed as per the remarks preceding this lemma. Therefore, we may write ˆ

|∂ P | = −1 e , − µ t t f (y) t Wt2 (y z)[ f (z) fB(x,ηt)] d (z) Rn and a change of variables implies that ˆ

|∂ P | = −1 η e , − µ . sup t t f (y) sup t W(t/η)2 (y z)[ f (z) fB(x,t)] d (z) (y,t)∈Γη(x) (y,t)∈Γ(x) Rn We can then obtain (3.8) by following the proof of (3.7) with α = 1 in order to / show that this is bounded by [Mµ(|∇ f |p)(x)]1 p, since the doubling property of µ ensures that ( ) | − |2 e C1,η y z |ηW /η 2 (y, z)| ≤ exp −C ,η (t ) µ(B(y, t)) 2 t2 for some positive constants C1,η and C2,η that depend on η. We obtain (3.9) as an −1 immediate consequence of (3.8) and the fact that η ∂tPηt = (∂sPs)|s=ηt.

To prove (3.10), let η > 0, set uηt := Pηt f and choose a non-negative function Φ ∈ ∞ , η Φ ≡ , η |∇ Φ| ≲ η −1 > Cc (B(y 2 t)) such that 1 on B(y t) and x ( t) . Let c 0 denote a constant that will be chosen later. The deﬁnition of Lµ implies that ˆ 2 1 2 2 |∇xPηt f | dµ ≤ |∇xuηt| Φ dµ ,η µ(B(y, ηt)) Rn B(y t) ˆ ≲ 1 ⟨A∇ , ∇ − ⟩Φ2 µ , η xuηt x(uηt c) (B(y t)) ˆRn { } = 1 ⟨A∇ , ∇ − Φ2 ⟩− ⟨A∇ , ∇ Φ − ⟩Φ µ , η xuηt x[(uηt c) ] 2 xuηt x (uηt c) (B(y t)) ˆRn { } ≲ 1 L − Φ2 + |∇ ||∇ Φ|| − Φ| µ µ , η ( µuηt)(uηt c) xuηt x (uηt c) d (B(y t)) ˆRn ( ) 1 1 2 ≤ |∂ η || η − |Φ + |∇ η ||∇ Φ|| η − |Φ µ 2 tu t u t c xu t x u t c d µ(B(y, ηt)) B(y,2ηt) 2η t CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 15

=: I + II.

Now ﬁx c := f˜B(x,3ηt). To estimate I, we use Cauchy’s inequality and the doubling property of µ, combined with the fact that B(x, ηt) ⊆ B(y, 2ηt) ⊆ B(x, 3ηt), to obtain

2 −1 η 2 2 2 |∇xPηt f | dµ ≲ I + II ≲ Mµ(|η N∗ (∂tPηt f )| + |N∗(∂tPt f )| + |∇ f | )(x), B(y,ηt) which, combined with (3.8) and (3.9), implies (3.10). □

The pointwise estimates in Lemma 3.6 have the following corollary. Corollary 3.11. Let n ≥ 2 and suppose that A ∈ E(n, λ, Λ, µ) for some constants n 0 < λ ≤ Λ < ∞ and an A2-weight µ on R . If η > 0, then ( ) µ { ∈ Rn | η ∂ P | > κ} ≲ κ−2∥∇ ∥2 , (3.12) x : N∗ ( t t f )(x) η f 2 Rn,Rn ( ) Lµ( ) µ { ∈ Rn |η−1 η ∂ P | > κ} ≲ κ−2∥∇ ∥2 , (3.13) x : N∗ ( t ηt f )(x) f L2(Rn,Rn) ( ) µ n eη −2 2 (3.14) µ {x ∈ R : |N (∇ Pη f )(x)| > κ} ≲ κ ∥∇ f ∥ n n , ∗,µ x t Lµ2(R ,R ) 1,2 n for all κ > 0 and f ∈ Wµ (R ), where the implicit constants depend only on n, λ, Λ µ η and [ ]A2 , as well as on in (3.12).

Proof. Estimates (3.12) and (3.13) follow respectively from (3.8) and (3.9), in the case p = 2, since Mµ satisﬁes the weak-type estimate in (3.1). To prove (3.14), note n that there exists 1 < q < 2 such that µ is an Aq-weight on R (see, for instance, Section 3 in Chapter V of [S2]). Therefore, combining (3.10) in the case p = q with (3.1) and noting that 2/q > 1, we obtain ( ) ( ) µ { ∈ Rn | eη ∇ P | > κ} ≲ κ−2 ∥ |∇ |q ∥2/q + ∥∇ ∥2 x : N∗,µ( x ηt f )(x) Mµ( f ) 2/q f 2 Rn,Rn Lµ (Rn) Lµ( ) −2 2 ≲ κ ∥∇ f ∥ n n Lµ2(R ,R ) κ > ∈ 1,2 Rn 1,2 Rn ⊆ 1,q Rn □ for all 0 and f Wµ ( ) (since Wµ ( ) Wµ,loc( )), as required. 16 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

4. The Carleson Measure Estimate

The purpose of this section is to prove the Carleson measure estimate (1.4) in Theorem 1.3. We adopt the strategy outlined at the end of Section 3.1 in [HKMP], although the crucial technical estimate, stated here as Theorem 4.10, is not at all an obvious extension of the uniformly elliptic case. Moreover, establishing the Carleson measure estimate directly allows us to avoid “good-λ” inequalities and 1,2 thus apply a change of variables based on the W0,µ-Hodge decomposition in (2.10), 1,2+ϵ ffi ϵ > instead of the W0 -version (for a su ciently small 0) required in [HKMP]. The technical result in Theorem 4.10 establishes (1.4) on certain “big pieces” of all cubes. The passage to the general estimate ultimately follows from the self- improvement property for Carleson measures in Lemma 2.22. This requires, however, that the Carleson measure estimate on the full gradient ∇u of a solution u can be controlled by the same estimate on its transversal derivate ∂tu, which is the con- tent of Lemma 4.2. We briefly postpone the statement and proof of Lemma 4.2 and Theorem 4.10, however, in order to deduce Theorem 1.3 from those results below. In contrast to the previous two sections, the results here concern solutions of the n+1 equation div(A∇u) = 0 in open sets Ω ⊆ R+ when n ≥ 2 and A is a t-independent coefficient matrix that satisfies (1.1) for some 0 < λ ≤ Λ < ∞ and an A2-weight µ on Rn. In particular, in Section 2, weighted Sobolev spaces were defined on open sets in Rd and matrix coefficients A ∈ E(d, λ, Λ, µ) were considered for all d ∈ N. Those results also hold here on open sets in the upper half-space with the weight µ(x, t):= µ(x) and the coefficients A(x, t):= A(x) for all (x, t) ∈ Rn+1, since then µ n+1 = µ Rn ∈ E + , λ, Λ, µ [ ]A2(R ) [ ]A2( ) and A (n 1 ). In particular, the solution space 1,2 Ω Wµ,loc( ) is defined and the regularity estimates in (2.16), (2.17) and (2.18) hold n+1 when Ω ⊆ R+ . We will also use, without reference, the well-known fact that if u is a solution of n+1 div(A∇u) = 0 in Ω ⊆ R+ , then ∂tu is also a solution in Ω. In particular, to see that ∂ 1,2 Ω Ω tu is in Wµ,loc( ), a Whitney decomposition of reduces matters to showing that ∂ 1,2 ⊂ Ω ℓ < 1 , ∂Ω tu is in Wµ (R) for all cubes R satisfying (R) 2 dist(R ). To this end, ff h = 1 + − ∈ define the di erence quotients Di u(X): h [u(X hei) u(X)] for all X R and n+1 h < dist(R, ∂Ω), where ei is the unit vector in the ith-coordinate direction in R . ffi h The t-independence of the coe cients implies that Dn+1u is a solution in R, so we use the identity Dh (∂ u) = ∂ (Dh u) and Caccioppoli’s inequality to obtain ¨ n+1 i ï n+1 ¨ | h ∂ |2 µ ≤ |∇ h |2 µ ≲ ℓ 2 | h |2 µ Dn+1( iu) d (Dn+1u) d (R) Dn+1u d R R ¨ 2R 2 2 ≤ ℓ(R) |∂tu| dµ =: K ∀h < dist(R, ∂Ω), 2R λ Λ µ where the implicit constant depends only on n, , and [ ]A2 , and the final bound 1,2 holds uniformly in h because u is in Wµ (R) (see Lemma 7.23 in [GT]). We can 1,2 then use Lemma 7.24 in [GT] to deduce that ∂tu is in Wµ (R) with the estimate ∥∂ ∂ u∥2 = ∥∂ ∂ u∥2 ≤ K for all i ∈ {1,..., n+1}, as required. Note that i t Lµ2(R) t i Lµ2(R) the proofs of Lemmas 7.23 and 7.24 in [GT] extend immediately to the weighted ∞ 1,2 context considered here because C (R) is still dense in Wµ (R). CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 17

Proof of Theorem 1.3 from Lemma 4.2 and Theorem 4.10. Let Q ⊂ Rn denote a ∞ n+1 n+1 cube and suppose that u ∈ L (R+ ) solves div(A∇u) = 0 in R+ . It follows a fortiori from Theorem 4.10 that there exist constants C, c0 > 0 and, for each cube ′ ′ ′ ′ ′ Q ⊆ Q, a measurable set F ⊂ Q such that µ(F ) ≥ c0µ(Q ) and ˆ ˆ l(Q′) 1 | ∂ , |2 µ dt ≤ ∥ ∥2 , ′ t tu(x t) d (x) C u ∞ µ(Q ) 0 F′ t λ Λ µ where C and c0 depend only on n, , and [ ]A2 . The coeﬃcient matrix A is t-independent, so ∂tu is also a solution and thus the degenerate version of Moser’s estimate in (2.16), followed by Caccioppoli’s inequality, shows that ∥t∂tu∥∞ ≲ ∥u∥∞. Moreover, the degenerate version of the de Giorgi–Nash Holder¨ regularity for solutions in (2.17) shows that ( )α ( )α |x − y| |x − y| |t∂ u(x, t) − t∂ u(y, t)| ≲ ∥t∂ u∥∞ ≲ ∥u∥∞ t t t t t for all x, y ∈ Q and t > 0, where all of the implicit constants and the exponent α > λ Λ µ 0 depend only on n, , and [ ]A2 . Therefore, we may apply Lemma 2.22 2 2 2 with {vt, α, β0, η, β} := {(t∂tu) , α, C∥u∥∞, c0, C∥u∥∞} to obtain ˆ ˆ ℓ(Q) 1 2 dt 2 (4.1) |t∂tu(x, t)| dµ(x) ≲ ∥u∥∞, µ(Q) 0 Q t λ Λ µ where the implicit constant depends only on n, , and [ ]A2 . This estimate holds for all cubes Q, so by Lemma 4.2, we conclude that (1.4) holds. □

We now dispense with the following lemma, which was used in the proof of Theorem 1.3 above to reduce to a Carleson measure estimate on the transversal derivative of solutions. The proof is adapted from Section 3.1 of [HKMP]. Lemma 4.2. Let n ≥ 2 and consider a cube Q ⊂ Rn. If A is a t-independent coefficient matrix that satisfies the degenerate bound and ellipticity in (1.1) for n some constants 0 < λ ≤ Λ < ∞ and an A2-weight µ on R , then any solution u ∈ L∞(4Q × (0, 4ℓ(Q))) of div(A∇u) = 0 in 4Q × (0, 4ℓ(Q)) satisfies ˆ ˆ ˆ ˆ ℓ(Q) 4ℓ(Q) 2 dt 2 dt 2 |t∇u(x, t)| dµ(x) ≲ |t∂tu(x, t)| dµ(x) + µ(Q)∥u∥∞, 0 Q t 0 4Q t λ Λ µ where the implicit constant depends only on n, , and [ ]A2 .

Proof. Let 0 < δ < 1/2 and set ΦQ(t):= Φ (t/ℓ(Q)), where Φ : R → [0, 1] denotes a C∞-function such that Φ(t) = 1 for all 2δ ≤ t ≤ 1 whilst Φ(t) = 0 for all t ≤ δ and t ≥ 2. Integrating by parts with respect to the t-variable and noting that ∥∂tΦ∥L∞([1,2]) ≲ 1 whilst ∥∂tΦ∥L∞([δ,2δ]) ≲ 1/δ, we obtain ˆ ˆ 2ℓ(Q) 2 I : = |∇u(x, t)| ΦQ(t)t dtdµ(x) ˆQ ˆ0 2ℓ(Q) ( ) 2 2 ≂ ∂t |∇u(x, t)| ΦQ(t) t dtdµ(x) ˆQ ˆ0 2ℓ(Q) 2 ≲ ⟨∇∂tu(x, t), ∇u(x, t)⟩ ΦQ(t)t dtdµ(x) Q 0 18 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS ˆ ˆ 2ℓ(Q) 2δℓ(Q) + |∇u(x, t)|2 t2dtdµ(x) + |∇u(x, t)|2 t2dtdµ(x) Q ℓ(Q) Q δℓ(Q) =: I′ + I′′ + I′′′. For the term I′, we apply Cauchy’s inequality with an arbitrary ϵ > 0, to obtain ˆ ˆ 2ℓ(Q) ′ 1 2 3 I ≤ ϵI + |∇∂tu(x, t)| t dtdµ(x). ϵ Q 0 For the term I′′, we apply Caccioppoli’s inequality, the doubling property of µ and the fact that t ≈ ℓ(Q) in the domain of the integration, to obtain ˆ ˆ 2ℓ(Q) I′′ ≂ ℓ(Q) |∇u(x, t)|2dtdµ(x) ℓ ˆQ ˆ(Q) 1 5ℓ(Q)/2 ≲ |u(x, t)|2dtdµ(x) ℓ(Q) 2Q ℓ(Q)/2 2 ≲ µ(Q)∥u∥∞.

′′′ ′′′ 2 For the term I , the same reasoning shows that I ≲ µ(Q)∥u∥∞. We now fix ϵ > 0, depending only on allowable constants, such that altogether ˆ ˆ 2ℓ(Q) 2 3 2 I ≲ |∇∂tu(x, t)| t dtdµ(x) + µ(Q)∥u∥∞, Q 0 which is justified since I < ∞ by Caccioppoli’s inequality and the support of ΦQ. To complete the estimate, we let {W j : j ∈ J} denote a collection of Whitney n+1 boxes∑ (from a Whitney decomposition of R+ ) such that W j ∩(Q × (0, 2ℓ(Q))) , Ø 1 , ≲ ffi ∂ and j∈J 2W j (x t) 1. The coe cient matrix A is t-independent, so tu is also a solution of div(A∇u) = 0 in each set W j, hence we may apply Caccioppoli’s inequality in combination with the fact that t ≂ l(W j) in W j, to obtain ˆ ˆ ¨ ℓ(Q) ∑ 2 dt 2 3 2 |t∇u(x, t)| dµ(x) ≲ |∇∂tu(x, t)| t dtdµ(x) + µ(Q)∥u∥∞ δℓ t 2 (Q) Q j∈J W j ∑ ¨ 2 2 ≲ l(W j) |∂tu(x, t)| dtdµ(x) + µ(Q)∥u∥∞ j∈J 2W j ˆ ˆ 4ℓ(Q) 2 dt 2 ≲ |t∂tu(x, t)| dµ(x) + µ(Q)∥u∥∞, 0 4Q t where the implicit constants do not depend on δ. The final result is then obtained by applying Fatou’s lemma to estimate the limit as δ approaches 0. □

The remainder of this section is dedicated to the proof of the crucial technical estimate, Theorem 4.10, that was used to prove Theorem 1.3. The proof adapts the change of variables from Section 3.2 of [HKMP] to the degenerate elliptic case. This is used to pull-back solutions to certain sawtooth domains where the Carleson measure estimate can be veriﬁed by reducing matters to the vertical square function estimates in Theorem 2.12, which we recall were obtained from the solution of the Kato problem in [CR3]. The following technical lemma, which reprises the CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 19

−t2Lµ 1 notation Pt := e for Lµ := − divµ( µ A∇) and A ∈ E(n, λ, Λ, µ) as in (2.6) and Lemma 3.6, will be used to justify these changes of variables. Lemma 4.3. Let n ≥ 2 and suppose that A ∈ E(n, λ, Λ, µ) for some constants n n 0 < λ ≤ Λ < ∞ and an A2-weight µ on R . Let Q ⊂ R denote a cube and n 1 ∞ suppose that f : 5Q → R is a measurable function such that µ f ∈ L (5Q). Let ϕ ∈ 1,2 A∇ϕ = κ > < η < / W0,µ(5Q) and suppose that div( ) div f in 5Q. If 0 0, 0 1 2 and x0 ∈ Q satisfy Λ(η, ϕ, A)(x0) ≤ κ0, where −1 η 2 1/2 (4.4) Λ(η, ϕ, A):= η N∗ (∂tPηtϕ) + N∗(∂tPtϕ) + [Mµ(|∇xϕ| )] + D∗,µϕ, then

(4.5) |∂tPηtϕ(x)| ≤ ηκ0 ∀(x, t) ∈ Γη(x0) and 1 (4.6) |(I − Pηt)ϕ(x)| ≲ η(κ0 + ∥ µ f∥∞)t ∀(x, t) ∈ Γη(x0) ∩ (2Q × (0, 4ℓ(Q))), λ Λ µ where the implicit constant depends only on n, , and [ ]A2 .

Proof. Suppose that κ0 > 0, 0 < η < 1/2 and x0 ∈ Q satisfy Λ(η, ϕ, A)(x0) ≤ κ0. It −1 η follows a fortiori that η N∗ (∂tPηtϕ)(x0) ≤ κ0, so (4.5) holds for all (x, t) ∈ Γη(x0). To prove (4.6), ﬁrst note that the properties of the semigroup imply that ˆ ηt

(4.7) |(I − Pηt)ϕ(x0)| = ∂sPsϕ(x0) ds ≤ ηtκ0 0 for all t > 0, sinceﬄ N∗(∂sPsϕ)(x0) ≤ κ0. Now let (x, t) ∈ Γη(x0) ∩ (2Q × (0, 4ℓ(Q))). We set ϕ ,η := ϕ(y)dy and apply estimate (3.7) with α = 2 to obtain x0 t B(x0,2ηt) |P ϕ − ϕ | ≲ η |∇ ϕ|2 1/2 ≤ η κ . (4.8) ηt( x0,ηt)(x) t[Mµ( x )(x0)] t 0

Next, since div(A∇(ϕ − ϕ(x0))) = div(A∇ϕ) = div f in 5Q, and since 0 < η < 1/2 ensures that B(x0, 2ηt) ⊆ 5Q, we may apply the degenerate version of Moser’s estimate for inhomogeneous equations in (2.20) to obtain ( ) 1/2 2 1 |ϕ(x) − ϕ(x0)| ≲ |ϕ(y) − ϕ(x0)| dµ(y) + ηt∥ µ f∥∞ B(x0,2ηt) (4.9) 1 ≲ ηt(D∗,µϕ(x0) + ∥ µ f∥∞) 1 ≲ ηt(κ0 + ∥ µ f∥∞). Combining estimates (4.7), (4.8) and (4.9), we obtain

|(I − Pηt)ϕ(x)| ≤ |ϕ(x) − ϕ(x0)| + |(I − Pηt)ϕ(x0)| + |P ϕ − ϕ | + |P ϕ − ϕ | ηt( x0,ηt)(x0) ηt( x0,ηt)(x) 1 ≲ η(κ0 + ∥ µ f∥∞)t, λ Λ µ □ which proves (4.6), as the implicit constant depends only on n, , and [ ]A2 . We now present the main technical result of this section. The proof is adapted from Section 3.2 of [HKMP], although some arguments have been simpliﬁed as detailed at the beginning of this section, and the additional justiﬁcation required in the degenerate elliptic case has been emphasized. 20 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

The strategy of the original proof in [HKMP] was motivated in-part by the fact that integration by parts is sufficient to establish the required estimate in the case when A has a certain block upper-triangular structure. A key idea in [HKMP] was to account for the presence of lower-triangular coefficients c (and upper-triangular ffi 1,2+ϵ coe cients) by decomposing them according to a W0 -Hodge decomposition. This was done locally on a given cube Q and the idea has been adapted here. First, 1,2 1 = µ − ∗∇φ the W0,µ-Hodge decomposition c 5Q h A|| is introduced in (4.13), where A|| is the n × n submatrix of A shown in (4.12). After integrating by parts, the divergence-free component µh provides valuable cancellation, whilst the adapted ∗ gradient vector field A|| ∇φ facilitates a reduction to the square function estimates in Theorem 2.12, which are implied by the solution to the Kato problem in [CR3], ∗ 1 ∗ for the boundary operator L||,µ := − divµ( µ A|| ∇x). − 2 ∗ ∗ ∗ = t L||,µ The latter estimates, however, require that L||,µ acts on the range of Pt : e and this is arranged by initially making the Dhalberg–Kenig–Stein-type pull-back 7→ − − ∗ φ ffi µ − ∗∇ ∗ φ t t (I Pηt) (x) so that the lower-triangular coe cients become h A|| xPηt . This change of variables is justified by choosing η > 0 small enough so that the pull-back is bi-Lipschitz in t. Once this is in place, a set F is introduced that contains a “big piece” of Q and on which the various maximal functions in Lemma 4.3 are bounded. The integration on F × (0, ℓ(Q)) is then performed by introducing a smooth test function Ψδ that equals 1 on F × (2δℓ(Q), 2ℓ(Q)) and is supported on a certain truncated sawtooth domain Ωη/8,Q,δ over F, where δ > 0 is an arbitrary (small) parameter that provides for a smooth truncation in the t-direction near the n+1 boundary of R+ . The main integration by parts is then performed in (4.32). The two principal terms S1 and S2 arise from the tangential and transversal integration by parts, respectively, where the former is taken with respect to the measure µ and thus requires additional justification from the uniformly elliptic case. These and numerous error terms are then shown to be appropriately under control.

Theorem 4.10. Let n ≥ 2 and consider a cube Q ⊂ Rn. If A is a t-independent coeﬃcient matrix that satisﬁes the degenerate bound and ellipticity in (1.1) for n some constants 0 < λ ≤ Λ < ∞ and an A2-weight µ on R , then for any solution u ∈ L∞(4Q × (0, 4ℓ(Q))) that solves div(A∇u) = 0 in 4Q × (0, 4ℓ(Q)), there exist constants C, c0 > 0 and a measurable set F ⊂ Q such that µ(F) ≥ c0µ(Q) and ˆ ˆ ℓ(Q) 1 2 dt 2 (4.11) |t∇u(x, t)| dµ(x) ≤ C∥u∥∞, µ(Q) 0 F t λ Λ µ where C and c0 depend only on n, , and [ ]A2 .

Proof. We begin by expressing the matrix A and its adjoint A∗ (which is just the transpose At, since the matrix coeﬃcients are real-valued) in the following form [ ] [ ] ∗ A|| b A c (4.12) A = , A∗ = || , ct d bt d where A|| denotes the n × n submatrix of A with entries (A||)i, j := Ai, j, 1 ≤ i, j ≤ n, t whilst b := (Ai,n+1)1≤i≤n is a column vector, c := (An+1, j)1≤ j≤n is a row vector and d := An+1,n+1 is a scalar. CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 21

Now consider a cube Q ⊂ Rn. The aim is to construct a set F ⊂ Q with the required properties. To this end, we apply the Hodge decomposition from (2.10) to 2 n the space Lµ(5Q, R ) in order to write

1 1 ∗ 1 1 e (4.13) µ c15Q = − µ A|| ∇φ + h, µ b15Q = − µ A||∇φe + h,

φ, φe ∈ 1,2 , e ∈ 2 , Rn = e = where Wµ,0(5Q) and h h Lµ(5Q ) are such that divµ h divµ h 0 and

( ) 2 c(x) (4.14) |∇φ(x)|2 + |h(x)|2 dµ(x) ≲ dµ(x) ≲ 1, 5Q 5Q µ

( ) 2 2 e 2 b(x) (4.15) |∇φe(x)| + |h(x)| dµ(x) ≲ dµ(x) ≲ 1. 5Q 5Q µ We extend each of φ, φ,e h, he to functions on Rn by setting them equal to 0 on Rn\5Q. 1 In Sections 2 and 3, we investigated the operators Lµ := − divµ( µ A∇) and −t2Lµ Pt := e for arbitrary coeﬃcient matrices A in E(n, λ, Λ, µ). We now set

1 −t2L||,µ L||,µ := − divµ( µ A||∇x), Pt := e , (4.16) ∗ ∗ ∗ − 2 ∗ = − 1 ∇ , = t L||,µ L||,µ : divµ( µ A|| x) Pt : e ∗ in order to apply those results in the cases A = A|| and A = A|| .

We now introduce two constants κ0, η > 0, which will be fixed shortly, and recall the function Λ(η, ϕ, A) from (4.4) to define the set F ⊂ Q by { ∗ (4.17) F := x ∈ Q : Λ(η, φ, A|| )(x) + Λ(η, φ,e A||)(x) } + eη ∇ ∗ φ + eη ∇ φe ≤ κ . N∗,µ( xPηt )(x) N∗,µ( xPηt )(x) 0 Applying the weak-type bounds in (3.1), (3.3), (3.13) and (3.14) followed by the estimates from the Hodge decomposition in (4.14) and (4.15), we obtain ( ) −2 2 2 −2 µ(Q \ F) ≲ κ ∥∇φ∥ n n + ∥∇φe∥ n n ≲ κ µ(Q), 0 Lµ2(R ,R ) Lµ2(R ,R ) 0 λ Λ µ where the implicit constants depend only on n, , and [ ]A2 . This allows us to now fix κ0 > 1 and some constant c0 > 0 such that µ(F) ≥ c0µ(Q), where both κ0 and c0 depend only on the allowed constants, and thus are independent of η. We now fix the value of η as follows. First, for 0 ≤ α ≤ 4 and β > 0, let ∪ ( ) Ωβ := Γβ(x), Ωβ,Q,α := Ωβ∩ 2Q×(αℓ(Q), 4ℓ(Q)) and Ωβ,Q := Ωβ,Q,0 x∈F n+1 denote the sawtooth domains in R+ spanned by cones centered on F of aperture β. Next, note that the properties of the Hodge decomposition in (4.13) imply that ∗ − div(A|| ∇φ) = div(c15Q) and − div(A||∇φe) = div(b15Q) in 5Q. Therefore, we now fix 0 < η < 1/2 in accordance with (4.5) and (4.6) such that { } |∂ ∗ φ |, |∂ φe | ≤ ηκ < / ∀ , ∈ Ω (4.18) max tPηt (x) tPηt (x) 0 1 8 (x t) η 22 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS and { } ∗ max |(I − P )φ(x)|, |(I − Pη )φe(x)| ( ηt { t }) (4.19) 1 1 ≲ η κ0 + max ∥ µ c∥∞, ∥ µ b∥∞ t ≲ ηκ0t < t/8 ∀(x, t) ∈ Ωη,Q, η λ Λ µ where and the implicit constants depend only on n, , and [ ]A2 . It remains to prove (4.11). We will achieve this by changing variables in the transversal direction using the mapping t 7→ τ(x, t), with x ∈ Rn fixed, defined by τ , = − − ∗ φ (x t): t (I Pηt) (x) and having Jacobian denoted by , = ∂ τ , = + ∂ ∗ φ . (4.20) J(x t): t (x t) 1 tPηt (x) In order to justify such changes of variables, we note from (4.18) and (4.19) that 7t 9t 7 9 (4.21) < τ(x, t) < and < J(x, t) < ∀(x, t) ∈ Ωη, . 8 8 8 8 Q In particular, for each x ∈ F and 0 ≤ α ≤ 1/8, this implies that the mapping t 7→ τ(x, t) is bi-Lipschitz in t on (2αℓ(Q), 2ℓ(Q)) with range ( ) (4.22) (4αℓ(Q), ℓ(Q)) ⊆ τ(x, ·) (2αℓ(Q), 2ℓ(Q)) ⊆ (αℓ(Q), 4ℓ(Q)). Moreover, for each 0 < β ≤ η, the mapping (x, t) 7→ ρ(x, t) defined by ρ , = , τ , = , + ∗ φ − φ (x t): (x (x t)) (x t Pηt (x) (x)) is bi-Lipschitz in t on Ωβ,Q with range

(4.23) Ω8β/9,Q ⊆ ρ(Ωβ,Q) ⊆ Ω8β/7,Q. Now consider a bounded solution u satisfying div(A∇u) = 0 in 4Q × (0, 4ℓ(Q)). ∞ The pull-back u := u ◦ ρ is in L (Ωη,Q) and div(A ∇u ) = 0 in Ωη,Q, where 1 [ 1 1 ] + ∇ φ − ∇ ∗ φ = JA|| b A|| x A|| xPηt A1 : µ − ∗∇ ∗ φ t ⟨ , ⟩/ ( h A|| xPηt ) Ap p J and [ ] [ ] ∇ τ(x, t) ∇ P∗ φ(x) − ∇ φ(x) (4.24) p(x, t):= x = x ηt x . −1 −1 ∇ = Ω Our statement that div(A1 u1) 0 in´ η,Q does not mean that A1 satisﬁes (1.1), 1,2 ∞ u ∈ W Ωη, n+1 ⟨A ∇u , ∇Φ⟩ = Φ ∈ C Ωη, only that 1 µ,loc( Q) and that R+ 1 1 0 for all c ( Q). To prove this, we combine the pointwise identity ⟨ ∇ ◦ ρ , ∇ ◦ ρ⟩ = ⟨ ∇ ◦ ρ , ∇ ◦ ρ ⟩ ∀ ∈ 1,2 ρ Ω (4.25) A (( u) ) ( v) J A1 (u ) (v ) v W0,µ( ( η,Q)) with the change of variables (x, t) 7→ ρ(x, t) on Ωη,Q, which is justiﬁed because ρ is bi-Lipschitz in t on Ωη,Q with range ρ(Ωη,Q) ⊂ 4Q × (0, 4ℓ(Q)) by (4.23). Also, we ∥1 ∥ ≤ ∥ ∥ note for later use that Ωη, u1 ∞ u ∞ and, using (4.21), that [ Q ]

∇xu1 − (∇xτ)(∂tu1)/J (4.26) |∇u1| ≲ + |∇xτ||∂tu1| = |(∇u) ◦ ρ| + |∇xτ| |∂tu1| (∂tu1)/J on Ωη,Q.

Next, in order to work with the pull-back solution u1, we consider an arbitrary constant 0 < δ ≤ 1/8 and deﬁne a smooth cut-oﬀ function Ψδ adapted to Ωη,Q CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 23

∞ as follows. Let δF(x):= dist(x, F), ﬁx a C -function Φ : R → [0, 1] satisfying Φ(t) = 1 when t < 1 and Φ(t) = 0 when t ≥ 1 , and then deﬁne ( 16 ) ( )( 8 ( )) δ F(x) t t n+1 Ψδ(x, t):= Φ Φ 1 − Φ ∀(x, t) ∈ R+ . ηt 32ℓ(Q) 16δℓ(Q) This function is designed so that Ψδ ≡ 1 on F ×(2δℓ(Q), 2ℓ(Q)), and since η < 1/2, we have supp Ψδ ⊆ Ωη/8,Q,δ and

1E (x, t) 1E (x, t) 1E (x, t) (4.27) |∇ , Ψδ(x, t)| ≲ 1 + 2 + 3 ∀(x, t) ∈ Ωη/ , ,δ, x t t ℓ(Q) δℓ(Q) 8 Q where { } E1 := (x, t) ∈ 2Q × (0, 4ℓ(Q)) : ηt/16 ≤ δF(x) ≤ ηt/8 ,

E2 := 2Q × (2ℓ(Q), 4ℓ(Q)),

E3 := 2Q × (δℓ(Q), 2δℓ(Q)).

In contrast to Section 3.2 in [HKMP], the cut-oﬀ function Ψδ introduced here incorporates an additional truncation in the t-direction at the boundary. This is done to simplify subsequent integration by parts arguments, since it ensures that n+1 Ψδ vanishes on the boundary of R+ . For later purposes, it is also convenient to isolate the following general fact here. ∈ Z Dη ′ ⊂ Rn Remark 4.28. For each k , let k denote the grid of dyadic cubes Q −k ′ −k such that η2 /64 ≤ diam Q < η2 /32. If C0 > 0 and (vt)t>0 is a collection of non-negative measurable functions such that

µ ≤ ∀ ∈ Z, ∀ ′ ∈ Dη, sup vt(x) d (x) C0 k Q k t∈[2−k,2−k+1] Q′ then ¨ ( ) 1 , 1 , 1 , E1 (x t) E2 (x t) E3 (x t) (4.29) + + vt(x) dµ(x)dt ≲ C0µ(Q), n+1 ℓ δℓ R+ t (Q) (Q) λ Λ µ where the implicit constant depends only on n, , and [ ]A2 . To see this, ﬁrst observe that since δ is a Lipschitz mapping with constant 1, we have F { } ηt Q(1) × [2−k, 2−k+1] ⊆ Ee := (x, t) ∈ 4Q × (0, 4ℓ(Q)) : ≤ δ (x) ≤ Cηt , 1 C F Q(2) × [2−k, 2−k+1] ⊆ 4Q × (ℓ(Q), 8ℓ(Q)), Q(3) × [2−k, 2−k+1] ⊆ 4Q × ((δ/2)ℓ(Q), 4δℓ(Q)) (i) −k −k+1 whenever Ei ∩ (Q × [2 , 2 ]) , Ø and i ∈ {1, 2, 3}. The estimate in (4.27) and the doubling property of µ then imply that the left side of (4.29) is bounded by   ˆ ˆ ∑ ∑ 2−k+1 8ℓ(Q) 4δℓ(Q)  dt  C 1 e dµ + C µ(Q) dt + C µ(Q) dt 0 E1 η 2−k Q′ t ℓ(Q) (δ/2)ℓ(Q) k∈Z Q′∈D ( k ) ˆ ˆ C δ η F (x) dt ≲ C0 dµ(x) + µ(Q) ≲ C0µ(Q), 1 δ t 4Q Cη F (x) as required. 24 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

We now proceed to prove (4.11). First, note that it suffices to show that ˆ ˆ ℓ(Q) 2 dt 2 (4.30) sup |t∇u(x, t)| dµ(x) ≲ ∥u∥∞µ(Q), 0<δ≤1/8 4δℓ(Q) F t since we may then obtain (4.11) by using Fatou’s lemma to pass to the limit as δ approaches 0. To this end, we use (4.22), followed by the bi-Lipschitz in t change of variables t 7→ τ(x, t) on (δℓ(Q), 2ℓ(Q)) for each x ∈ F, estimate (4.21) and identity (4.25), to obtain ˆ ˆ ˆ ˆ ℓ(Q) dt ℓ(Q) |t∇u(x, t)|2 dµ(x) ≲ ⟨A∇u, ∇u⟩ tdtdx δℓ t δℓ 4 (Q) F ˆF ˆ4 (Q) 2ℓ(Q) ≲ ⟨A1∇u1, ∇u1⟩ tdtdx ¨F 2δℓ(Q) 2 ≤ ⟨A1∇u1, ∇u1⟩Ψδ tdxdt. n+1 R+ Thus, in order to prove (4.30) and ultimately (4.11), it suffices to show that ¨ 2 2 (4.31) ⟨A1∇u1, ∇u1⟩Ψδ tdxdt ≲ ∥u∥∞µ(Q) ∀0 < δ ≤ 1/8, n+1 R+ λ Λ µ where the implicit constant depends only on n, , and [ ]A2 . ∇ = Ω Ψ2 ∈ 1,2 Ω Next, we recall that div(A1 u1) 0 in η,Q, noting that u1 δt W0,µ( η,Q), and then integrate by parts to obtain ¨ ¨ ⟨ ∇ , ∇ ⟩Ψ2 = −1 ⟨ ∇ 2 , ∇ Ψ2 ⟩ A1 u1 u1 δ tdxdt A1 (u1) ( δt) dxdt Rn+1 2 Rn+1 + ¨ + ¨ = −1 ⟨∇ 2 , 1 ∗ ⟩Ψ2 µ − 1 ⟨ ∇ 2 , ∇ Ψ2 ⟩ (u1) µ A1en+1 δ d dt A1 (u1) ( δ) tdxdt 2 Rn+1 2 Rn+1 ¨ + ¨ + (4.32) = 1 2 ∗ ∗ φ Ψ2 µ + 1 2∂ ⟨ , ⟩/ Ψ2 u1(L||,µPηt ) δ d dt u1 t( Ap p J) δ dxdt 2 Rn+1 2 Rn+1 ¨+ + ¨ − 1 ⟨ ∇ 2 , ∇ Ψ2 ⟩ + 1 2⟨ , ∇ Ψ2 ⟩ A1 (u1) ( δ) tdxdt u1 en+1 A1 ( δ) dxdt n+1 n+1 2 R+ 2 R+ =: S1 + S2 + E1 + E2, where en+1 := (0, ..., 0, 1) denotes the unit vector in the t-direction. In particular, note that the tangential integration by parts ˆ ˆ ⟨∇ 2 , − 1 ∗∇ ∗ φ⟩Ψ2 µ = 2 − 1 ∗∇ ∗ φ Ψ2 µ, x(u1) h µ A|| xPηt δ d u1 divµ[(h µ A|| xPηt ) δ] d Rn Rn with respect to the measure µ, is justified by the definition of the operator divµ, ∗ φ ∈ ∗ = − 1 ∗∇ ∗ φ Ψ2 ∈ since Pηt Dom(L||,µ) and divµ h 0 imply that (h µ A|| xPηt ) δ Dom(divµ) (recall (2.8), (2.9) and (4.16)). Meanwhile, the transversal integration by parts ˆ ∞ ˆ ∞ ∂ 2 ⟨ , ⟩/ Ψ2 = − 2∂ ⟨ , ⟩/ Ψ2 t(u1)( Ap p J) δ dt u1 t[( Ap p J) δ] dt 0 0 n+1 is justified because Ψδ vanishes on the boundary of R+ . CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 25

We proceed to prove that, for all σ ∈ (0, 1), each term in (4.32) is controlled by ¨ 2 −1 2 (4.33) S1 + S2 + E1 + E2 ≲ σ ⟨A1∇u1, ∇u1⟩Ψδ tdxdt + σ ∥u∥∞µ(Q), n+1 R+ λ Λ µ where the implicit constant depends only on n, , and [ ]A2 . Estimate (4.31) will then follow by fixing a sufficiently small σ ∈ (0, 1), depending only on allowed constants, to move the integral in (4.33) to the left side of (4.32). This is justified because the integral in (4.33) is finite by Caccioppoli’s inequality and the fact that n+1 Ψδ vanishes in a neighbourhood of the boundary of R+ (supp Ψδ ⊆ Ωη/8,Q,δ). We now prove (4.33) in three steps to complete the proof.

Step 1: Estimates for the error terms E1 and E2 in (4.32). We ﬁrst apply Cauchy’s inequality with σ to write ¨

≤ 1 ⟨ ∇ 2 , ∇ Ψ2 ⟩ E1 A1 (u1) ( δ) tdxdt 2 Rn+1 ¨ +

= 2 ⟨A1∇u1, ∇Ψδ⟩u1Ψδ tdxdt n+1 ¨ R+ ¨ ≲ σ ⟨ ∇ , ∇ ⟩Ψ2 + σ−1 2⟨ ∇Ψ , ∇Ψ ⟩ A1 u1 u1 δ tdxdt u1 A1 δ δ tdxdt n+1 n+1 ¨R+ R+ = σ ⟨ ∇ , ∇ ⟩Ψ2 + σ−1 ′ . : A1 u1 u1 δ tdxdt E1 n+1 R+ ∗ We then use µh = c15Q + A|| ∇φ from (4.13), the degenerate bound in (1.1) for A, ∥1 ∥ ≲ ∥ ∥ ∇Ψ the bound Ωη,Q u1 ∞ u ∞ and the estimate for δ from (4.27) to obtain ¨ ( ) 1 1 1 ( ) ′ + ≲ ∥ ∥2 E1 + E2 + E3 + |∇ − ∗ φ|2 µ , E1 E2 u ∞ ℓ δℓ 1 x(I Pηt) d dt Ωη/8,Q t (Q) (Q) 2 2 where (4.27) ensures that |∇(Ψδ)| and |∇Ψδ| t can be controlled in the same manner. = 1 +|∇ − ∗ φ|2 In order to apply Remark 4.28 with vt Ωη/8,Q (1 x(I Pηt) ), we observe that ∈ Z ′ ∈ Dη Ω ∩ ′ × −k, −k+1 , ∈ if k , Q k and η/8,Q,δ (Q [2 2 ]) Ø, then there exists x0 F ′ −k ′ n such that Q ⊆ ∆(x0, η2 ) ⊆ CQ , where ∆ is used to denote balls in R , hence ′ −k −k+1 (4.34) Q × [2 , 2 ] ⊆ Ωη,2Q,δ/4 and the doubling property of µ implies that

|∇ − ∗ φ|2 µ ≲ |∇ ∗ φ|2 µ + |∇ φ|2 µ (4.35) x(I Pηt) d xPηt d x d Q′ ∆ x ,ηt ∆ x ,η −k [ ] ( 0 ) ( 0 2 ) ≲ eη ∇ ∗ φ 2 + |∇ φ|2 ≲ κ2 ≲ ∀ ∈ −k, −k+1 , N∗,µ( xPηt )(x0) Mµ( x )(x0) 0 1 t [2 2 ] where in the last line we used the deﬁnition of the set F in (4.17) and the weighted e maximal operators N∗,µ and Mµ from Section 3. It thus follows from (4.29) that ′ 2 E + E ≲ ∥u∥∞µ(Q), so altogether we have 1 2 ¨ 2 −1 2 (4.36) E1 + E2 ≲ σ ⟨A1∇u1, ∇u1⟩Ψδ tdxdt + σ ∥u∥∞µ(Q) ∀σ ∈ (0, 1). n+1 R+

Step 2: Estimates for the term S1 in (4.32). 26 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

∗ 2 ∗ ∗ 2 n We note that ∂tPη = −2η tL||,µPη on Lµ(R ) and integrate by parts in t to write ¨ t t = 1 2 ∗ ∗ φ Ψ2 µ S1 u1(L||,µPηt ) δ d dt 2 Rn+1 ¨+ ¨ = −1 2∂ ∗ ∗ φ Ψ2 µ + 1 ∂ ∂ ∗ φ Ψ2 µ u1 t(L||,µPηt ) δ td dt (u1 tu1)( tPηt ) δ d dt 2 Rn+1 2η2 Rn+1 ¨+ + + 1 2 ∂ ∗ φ Ψ ∂ Ψ µ = ′ + ′′ + ′′′, u1( tPηt ) δ t δ d dt : S1 S1 S1 η2 n+1 2 R+ n+1 where there is no boundary term because Ψδ vanishes on the boundary of R+ . ′′′ To estimate S1 , we use the deﬁnition of the set F in (4.17), the estimate for |∇Ψ | ≡ δ from (4.27),¨ and Remark 4.28 in the case vt 1, to obtain ′′′ ≲ ∥ ∥2 η ∂ ∗ φ |∂ Ψ | µ ≲ ηκ ∥ ∥2 µ ≲ ∥ ∥2 µ . S1 u ∞ N∗ ( tPηt ) t δ d dt 0 u ∞ (Q) u ∞ (Q) Ωη/8,Q ′ ∂ ∗ ∗ φ = ∗ ∂ ∗ φ φ ∈ 1,2 Rn To estimate S1, we observe that t(L||,µPηt ) L||,µ( tPηt ), since Wµ,0( ) ∂ ∗ = − η2 ∗ ∗ ∗ 1,2 Rn and tPηt 2 tPηtL||,µ on the dense subset Dom(L||,µ) of W0,µ( ) (note also that ∇ ∗ 2 t xPηt and hence its adjoint are bounded operators on Lµ, as can be seen from the proof of Theorem 2.12). We then apply Cauchy’s inequality with σ to write ¨

′ ≤ ∗ ∂ ∗ φ 2Ψ2 µ S1 L||,µ( tPηt )u1 δ td dt n+1 ¨R+

≲ ⟨ 1 ∗∇ ∂ ∗ φ , ∇ ⟩ Ψ2 µ µ A|| x( tPηt ) xu1 u1 δ td dt n+1 ¨R+

+ ⟨ 1 ∗∇ ∂ ∗ φ , ∇ Ψ ⟩ 2Ψ µ = + (4.37) µ A|| x( tPηt ) x δ u1 δ td dt : J K n+1 ¨ R+ ¨ ≲ σ |∇ |2 Ψ2 µ + σ−1 + 2|∇ ∂ ∗ φ|2Ψ2 µ xu1 δ td dt ( 1) u1 x tPηt δ td dt n+1 n+1 ¨R+ R+ + 2 |∇ Ψ |2 µ = σ ′ + σ−1 + ′ + ′ , u1 x δ td dt : S11 ( 1)S12 S13 n+1 R+ where the integration by parts in x, with respect to the measure µ, is justiﬁed by the ∗ deﬁnition of the operator L||,µ (recall (2.8), (2.9) and (4.16)). The terms J and K are highlighted above for reference in Step 3. ′ |∇Ψ | To estimate S13, we use the estimate for δ from (4.27) and Remark 4.28 in ≡ ′ ≲ ∥ ∥2 µ the case vt 1, to obtain S13 u ∞ (Q). ′ ∇ ∂ ∗ = − η2 ∇ ∗ ∗ 2 Rn To estimate S12, we observe that x tPηt 2 t xL||,µPηt on Lµ( ) and then 1,2 apply the vertical square function estimate from (2.14) followed by the W0,µ(5Q)- Hodge estimate for φ from (4.14) to obtain ¨ ¨ ′ ≲ 2|∇ ∂ ∗ φ|2Ψ2 µ ≲ ∥ ∥2 | 2∇ ∗ ∗ φ|2 µdt S12 u1 x tPηt δ td dt u ∞ t xL||,µPηt d n+1 n+1 R+ R+ t 2 2 2 ≲ ∥u∥ ∥∇φ∥ n n ≲ ∥u∥ µ(Q). ∞ Lµ2(R ,R ) ∞ ′ ′′ The terms S11 and S1 will now be estimated together. We again apply Cauchy’s inequality with σ, followed by the vertical square function estimate from (2.13) CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 27

∗ 1,2 with Lµ = L||,µ and the W ,µ(5Q)-Hodge estimate for φ from (4.14) to obtain ¨ 0 ¨

σ ′ + ′′ ≲ σ |∇ |2 Ψ2 µ + ∂ ∂ ∗ φ Ψ2 µ S11 S1 xu1 δ td dt (u1 tu1)( tPηt ) δ d dt n+1 n+1 ¨ R+ ¨ R+ ≲ σ |∇ |2Ψ2 µ + σ−1∥ ∥2 |∂ ∗ φ|2 µdt u1 δ td dt u ∞ tPηt d Rn+1 Rn+1 t ¨ + ¨ + 2 2 2 2 −1 2 ≲ σ ⟨A1∇u1,∇u1⟩Ψδ tdxdt + σ |∇xτ| |∂tu1| Ψδ tdµdt + σ ∥u∥∞µ(Q), n+1 n+1 R+ R+ where we combined the pointwise estimates for ∇u1 and J from (4.26) and (4.21) with identity (4.25) and the ellipticity of A to deduce the ﬁnal inequality. We use the dyadic decomposition from Remark 4.28 to write ¨ ˆ ˆ ∑ ∑ 2−k+1 |∇ τ|2|∂ |2Ψ2 µ ≤ 1 |∇ τ|2|∂ |2 µ . (4.38) x tu1 δ td dt Ωη,Q,δ x tu1 td dt Rn+1 −k ′ + k∈Z ′∈Dη 2 Q Q k ∈ Z ′ ∈ Dη Ω ∩ ′ × −k, −k+1 , Observe that if k , Q k and η/8,Q,δ (Q [2 2 ]) Ø, then as in ′ − − + (4.34) and (4.35), it holds that Q × [2 k, 2 k 1] ⊆ Ωη, ,δ/ and 2Q 4 |∇ τ , |2 µ ≲ κ2 ∀ ∈ −k, −k+1 . x (x t) d (x) 0 t [2 2 ] Q′ 7 < τ , < 9 , ≂ ′ × −k, −k+1 Also, we have 8 t (x t) 8 t and J(x t) 1 on Q [2 2 ] by (4.21), so the degenerate version of Moser’s estimate in (2.16) and t-independence show that

2t 2 2 2 sup |∂tu1(x, t)| = sup |J(x, t)∂τu(x, τ(x, t))| ≲ |∂su(y, s)| dsdµ(y) x∈Q′ x∈Q′ 2Q′ t/2 for all t ∈ [2−k, 2−k+1]. In particular, note that { } η δ ′ −k−1 −k+2 ∗ n+1 5 s 2Q × [2 , 2 ] ⊆ Ω := (y, s) ∈ R+ : δ (y) < , ℓ(Q) < s < 8ℓ(Q) , F 8 2 , ∈ ′ × −k, −k+1 δ < 1 η since there exists (x0 t0) Q [2 2 ] satisfying F(x0) 8 t0, whence 1 5 5 δ (y) < diam(2Q′) + ηt ≤ η2−k ≤ ηs ∀y ∈ 2Q′ and s ≥ 2−k−1, F 8 0 16 8 −k −k+1 whilst δℓ(Q) < t0 < 4ℓ(Q) implies that [2 , 2 ] ⊆ ((δ/2)ℓ(Q), 8ℓ(Q)). The observations in the preceding paragraph show that (4.38) is bound by ˆ ( )(ˆ ˆ ) ∑ ∑ 2−k+1 2t 2 2 |∇xτ| dµ |∂su(y, s)| 1Ω∗ (y, s) dsdµ(y) dt −k ′ ′ / k∈Z ′∈Dη 2 Q 2Q t 2 Q k ˆ ˆ ∑ ∑ 2−k+2 2 ≲ |∂su(y, s)| 1Ω∗ (y, s) sdµ(y)ds η 2−k−1 2Q′ k∈Z Q′∈D (¨ k ¨ ) 2 2 ≲ |∂su(y, s)| sdµ(y)ds + |∂su(y, s)| sdµ(y)ds := M + E, Ω∗∗ Ω∗\Ω∗∗ ∑ ∑ η 1 ≲ 1 where we used the fact that k∈Z Q′∈D 2Q′×[2−k−1,2−k+2] Rn+1 and introduced { k + } ∗∗ n+1 Ω := (y, s) ∈ R+ : δF(y) < ηs/18, 4δℓ(Q) < s < ℓ(Q) . 28 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

To estimate the main term M, we use (4.21)-(4.23) to observe that

− ∗∗ ρ 1(Ω ) ⊆ Ω η ∩ (2Q × (2δℓ(Q), 2ℓ(Q))). 16

Thus, since Ψδ ≡ 1 on these sets, the change of variables (y, s) 7→ ρ(y, s) gives ¨ ¨ 2 2 2 M ≲ |(∂tu) ◦ ρ| J Ψδ tdµdt ≲ ⟨A1∇u1, ∇u1⟩Ψδ tdxdt, n+1 n+1 R+ R+ where we used identity (4.25) and the ellipticity of A to deduce the ﬁnal inequality. To estimate the error term E, recall that the degenerate version of Moser’s estimate in (2.16), followed by Caccioppoli’s inequality, ensures that ∥s∂su∥∞ ≲ ∥u∥∞. Thus, by the deﬁnition of Ω∗ \ Ω∗∗ and the doubling property of µ, we obtain ( ) ˆ ˆ 18 ˆ ˆ η δF (y) 8ℓ(Q) 4δℓ(Q) 2 ds ds ds 2 E ≲ ∥u∥∞ + + dµ(y) ≲ ∥u∥∞µ(Q). 8 δ s ℓ s δ/ ℓ s 2Q 5η F (y) (Q) ( 2) (Q) ˜ ′ ′′ 2 −1 2 σS + S ≲ σ n+1 ⟨A ∇u , ∇u ⟩Ψ tdxdt + σ ∥u∥ µ Q This shows that 11 1 R+ 1 1 1 δ ∞ ( ), hence ¨ 2 −1 2 (4.39) S1 ≲ σ ⟨A1∇u1, ∇u1⟩Ψδ tdxdt + σ ∥u∥∞µ(Q) ∀σ ∈ (0, 1). n+1 R+

Step 3: Estimates for the term S2 in (4.32). We observe that since A is t-independent it is possible to write ¨ = 2∂ ⟨ , ⟩/ Ψ2 2S2 u1 t( Ap p J) δ dxdt n+1 ¨R+ ¨ = 2∂ / ⟨ , ⟩Ψ2 + 2/ ⟨∂ , ∗ ⟩Ψ2 u1 t(1 J) Ap p δ dxdt (u1 J) tp A p δ dxdt n+1 n+1 ¨R+ R+ + 2/ ⟨ , ∂ ⟩Ψ2 = + + . (u1 J) Ap tp δ dxdt : I II III n+1 R+

, = +∂ ∗ φ To estimate I, we recall the Jacobian J(x t) 1 tPηt (x) from (4.20) and then integrate by parts in t to write ¨ ∂2P∗ φ = − 2 t ηt ⟨ , ⟩Ψ2 I u1 Ap p δ dxdt Rn+1 J2 ¨ + ¨ ∂ P∗ φ ∂ P∗ φ = ∂ 2 t ηt ⟨ , ⟩Ψ2 + 2 t ηt ∂ ⟨ , ⟩ Ψ2 t(u1) Ap p δ dxdt u1 t( Ap p ) δ dxdt Rn+1 J2 Rn+1 J2 ¨+ + ¨ ∂ P∗ φ + 2∂ ∗ φ ∂ −2 ⟨ , ⟩Ψ2 + 2 t ηt ⟨ , ⟩∂ Ψ2 u1 tPηt t(J ) Ap p δ dxdt u1 Ap p t( δ) dxdt n+1 n+1 2 R+ R+ J =: I1 + I2 + I3 + I4,

n+1 where there is no boundary term because Ψδ vanishes on the boundary of R+ . CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 29

To estimate I1, we recall that J ≂ 1 on supp Ψδ ⊆ Ωη/8,Q,δ by (4.21) and then apply Cauchy’s inequality with σ to obtain ¨ 2 2 2 |I1| ≲ σ |∂tu1| |p| Ψδ tdµdt Rn+1 (4.40) +¨ + σ−1 2|∂ φ|2| |2Ψ2 µdt = σ ′ + σ−1 ′′. u1 tPηt p δ d : I1 I1 n+1 R+ t

′ | |2 = + |∇ τ|2 To estimate I1, recall that p 1 x by the deﬁnition of p in (4.24), so we follow the treatment of (4.38) above to obtain ¨ ′ ≲ ⟨ ∇ , ∇ ⟩Ψ2 + ∥ ∥2 µ . I1 A1 u1 u1 δ tdxdt u ∞ (Q) n+1 R+ ′′ ∥1 ∥ ≲ ∥ ∥ To estimate I1 , recall that Ωη,Q u1 ∞ u ∞ and use the dyadic decomposition from Remark 4.28 to obtain ˆ ˆ ∑ ∑ 2−k+1 ′′ ≲ ∥ ∥2 ∥∂ ∗ φ∥2 | |2Ψ2 µdt I1 u ∞ tPηt L∞(Q′×[2−k,2−k+1]) p δ d η 2−k Q′ t k∈Z Q′∈D ∑ ∑k ≲ ∥ ∥2 µ ′ ∥∂ ∗ φ∥2 u ∞ (Q ) tPηt L∞(Q′×[2−k,2−k+1]) k∈Z ′∈Dη Q k

∑ ∑ 2−k+2 ≲ ∥ ∥2 µ ′ |∂ ∗ φ|2 µ u ∞ (Q ) tPηt d dt −k−1 ′ (4.41) k∈Z ′∈Dη 2 2Q Q k ˆ ˆ ∑ ∑ 2−k+2 ≲ ∥ ∥2 |∂ ∗ φ|2 µdt u ∞ tPηt d η 2−k−1 2Q′ t k∈Z Q′∈D ¨ k 2 ∗ 2 ∗ −t L||,µ 2 dt ≲ ∥u∥∞ |tL||,µe φ| dµ n+1 R+ t 2 2 2 ≲ ∥u∥ ∥∇φ∥ n n ≲ ∥u∥ µ(Q), ∞ Lµ2(R ,R ) ∞

| |2Ψ2 ≤ 1 +|∇ − ∗ φ|2 where the second line uses the pointwise bound p δ Ωη/8,Q,δ (1 x(I Pηt) ) and estimate (4.35), the third line uses the parabolic version of the degenerate ∗ −tL||,µ Moser-type estimate in (2.16) (see Theorem B in [F]), noting that v := ∂t(e φ) ∂ = − ∗ |∂ ∗ φ | ≲ | , η2 2 | solves tv L||,µv whilst tPηt (x) t v(x t ) , and the ﬁnal line uses the ver- L = ∗ 1,2 tical square function estimate from (2.13) with µ L||,µ and the W0,µ(5Q)-Hodge estimate for φ from (4.14).

To estimate I2, we again use the bound J ≂ 1 on supp Ψδ ⊆ Ωη/8,Q,δ from (4.21), = ∇ ∗ − φ, − and then recall the deﬁnition p : ( x(Pηt I) 1) from (4.24) to obtain ¨ ¨ | | ≲ 2|∇ ∂ ∗ φ|2Ψ2 µ + 2|∂ ∗ φ|2| |2Ψ2 µdt. (4.42) I2 u1 x tPηt δ td dt u1 tPηt p δ d n+1 n+1 R+ R+ t ′ The ﬁrst integral in (4.42) is the same as S12 from (4.37) whilst the second integral ′′ | | ≲ ∥ ∥2 µ is the same as I1 from (4.40), hence I2 u ∞ (Q). 30 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

|∂ ∗ φ| < / To estimate I3, we use the bound tPηt 1 8 guaranteed by (4.18) to deduce |∂ −2 | = |∂ + ∂ ∗ φ −2| ≲ |∂2 ∗ φ| Ψ ⊆ Ω that t(J ) t(1 tPηt ) t Pηt on supp δ η/8,Q,δ and write ¨ ¨ | | ≲ 2|∂ ∗ φ|2| |2Ψ2 µdt + 2|∂2 ∗ φ|2| |2Ψ2 µ = ′ + ′′ I3 u1 tPηt p δ d u1 t Pηt p δ td dt : I3 I3 n+1 n+1 R+ t R+ ′ ′′ ′ ≲ ∥ ∥2 µ To estimate I3, we note that it is the same as I1 from (4.40), thus I3 u ∞ (Q). ′′ To estimate I3 , we follow the estimates and justiﬁcation provided for (4.41), − ∗ ∂ = ∂2 tL||,µ φ ∂ ∂ = − ∗ ∂ noting in addition that tv t (e ) solves t( tv) L||,µ( tv), to obtain ˆ ˆ ∑ ∑ 2−k+1 ′′ ≲ ∥ ∥2 ∥ ∂2 ∗ φ∥2 | |2Ψ2 µdt I3 u ∞ t t Pηt L∞(Q′×[2−k,2−k+1]) p δ d η 2−k Q′ t k∈Z Q′∈D ∑ ∑k ≲ ∥ ∥2 µ ′ ∥ | ∗ ∗ φ| + | 2∂ ∗ ∗ φ | ∥2 u ∞ (Q ) tL||,µPηt t t(L||,µPηt ) L∞(Q′×[2−k,2−k+1]) k∈Z ′∈Dη Q k

∑ ∑ 2−k+2 ≲ ∥ ∥2 µ ′ | ∗ ∗ φ|2 + | 2∂ ∗ ∗ φ |2 µ u ∞ (Q ) ( tL||,µPηt t t(L||,µPηt ) ) d dt η 2−k−1 2Q′ k∈Z Q′∈D ¨ k ¨ ≲ ∥ ∥2 | ∗ ∗ φ|2 µdt + ∥ ∥2 | 2∇ ∗ ∗ φ |2 µdt u ∞ tL||,µPηt d u ∞ t x,t(L||,µPηt ) d n+1 n+1 R+ t R+ t 2 2 2 ≲ ∥u∥ ∥∇φ∥ n n ≲ ∥u∥ µ(Q), ∞ Lµ2(R ,R ) ∞ |∂2 ∗ φ| ≲ |∂ ∗ ∗ φ | ≲ | ∗ ∗ φ| + | ∂ ∗ ∗ φ | where the second line uses t Pηt t(tL||,µPηt ) L||,µPηt t t(L||,µPηt ) , | ∗ ∗ φ | = | , η2 2 | |∂ ∗ ∗ φ | ≲ | ∂ , η2 2 | the third line uses L||,µPηt (x) v(x t ) and t(L||,µPηt )(x) t( tv)(x t ) , and the final line uses the vertical square function estimates from (2.13) and (2.14) ∗ 2 with Lµ = L||,µ, hence |I3| ≲ ∥u∥∞µ(Q) |∂ ∗ φ| ≲ ≂ | |2 ≤ + |∇ − ∗ φ|2 To estimate I4, we use tPηt 1, J 1 and p (1 x(I Pηt) ), which hold on supp Ψδ ⊆ Ωη/8,Q,δ by (4.18), (4.21) and (4.24), to reduce to the estimate ′ + | | ≲ ∥ ∥2 µ obtained for E1 E2, hence I4 u ∞ (Q). = ∇ ∗ − φ, − To estimate II, we use the definition p : ( x(Pηt I) 1) from (4.24) to note ∂ = ∇ ∂ ∗ φ, that tp ( x tPηt 0) and use the Hodge decomposition from (4.13) to write ⟨∂ , ∗ ⟩ = ⟨∇ ∂ ∗ φ, ∗∇ ∗ − φ − ⟩ = ⟨∇ ∂ ∗ φ, ∗∇ ∗ φ − µ ⟩ (4.43) tp A p x tPηt A|| x(Pηt I) c x tPηt A|| xPηt h for all x ∈ 5Q and t > 0. Using this and recalling that divµ h = 0, it follows that ¨ = 2/ ⟨∇ ∂ ∗ φ, ∗∇ ∗ φ − µ ⟩Ψ2 II (u1 J) x tPηt A|| xPηt h δ dxdt n+1 ¨R+ = 2/ ∂ ∗ φ ∗ ∗ φ Ψ2 µ (u1 J)( tPηt )(L||,µPηt ) δ d dt n+1 ¨R+ (4.44) − ∂ ∗ φ⟨∇ 2/ , ∗∇ ∗ φ − µ ⟩Ψ2 tPηt x(u1 J) A|| xPηt h δ dxdt n+1 ¨R+ − 2/ ∂ ∗ φ⟨∇ Ψ2 , ∗∇ ∗ φ − µ ⟩ (u1 J) tPηt x( δ) A|| xPηt h dxdt n+1 R+ =: II1 + II2 + II3, CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 31 where the integration by parts in x, with respect to the measure µ, is justified by the ∗ definition of the operator L||,µ (recall (2.8), (2.9) and (4.16)). ≂ ∗ ∗ φ = − η2 −1∂ ∗ φ To estimate II1, we use J 1 and L||,µPηt (2 t) tPηt to show that it ′′ | |2 | | ≲ ∥ ∥2 µ can be treated the same way as I1 in (4.40), without p , hence II1 u ∞ (Q). ≂ |∇ −1 | = |∇ + ∂ ∗ φ −1| ≲ |∇ ∂ ∗ φ| To estimate II2, we use J 1, x(J ) x(1 tPηt ) x tPηt and apply Cauchy’s inequality inequality with σ to obtain ¨ ¨ | | ≲ σ |∇ |2Ψ2 µ + 2|∇ ∂ ∗ φ|2Ψ2 µ II2 xu1 δ td dt u1 x tPηt δ td dt Rn+1 Rn+1 (4.45) + ¨ + + σ−1 + 2|∂ ∗ φ|2 |∇ ∗ φ|2 + | |2 Ψ2 µdt. ( 1) u1 tPηt ( xPηt h ) δ d n+1 R+ t ′ The first integral is the same as S11 from (4.37) whilst the remaining two integrals |∇ ∗ φ|2 + | |2 | |2 are the same as those that bound I2 in (4.42), except ( xPηt h ) replaces p . This factor is controlled in the same way, however, since the Hodge decomposition 2 1 1 ∗ 2 2 in (4.13) implies that |h| = | µ c15Q + µ A|| ∇xφ| ≲ 1+|∇xφ| , so by (4.35) we obtain ¨ 2 −1 2 |II2| ≲ σ ⟨A1∇u1, ∇u1⟩Ψδ tdxdt + σ ∥u∥∞µ(Q). n+1 R+ To estimate II , we use J ≂ 1 and Cauchy’s inequality to write ¨ 3 ¨ | | ≲ 2|∇ Ψ |2 µ + 2|∂ ∗ φ|2 |∇ ∗ φ|2 + | |2 Ψ2 µdt II3 u1 x δ td dt u1 tPηt ( xPηt h ) δ d n+1 n+1 R+ R+ t ′ The first term above is the same as S13 in (4.37) whilst the remaining term is the 2 same as the last integral in (4.45), hence |II3| ≲ ∥u∥∞µ(Q). To estimate III, we observe by analogy with (4.43) that ⟨ , ∂ ⟩ = ⟨ ∇ ∗ − φ − , ∇ ∂ ∗ φ⟩ Ap tp A|| x(Pηt I) b x tPηt = ⟨ ∇ ∗ φ − φ + ∇ φe − µe, ∇ ∂ ∗ φ⟩ A|| x(Pηt ) A|| x h x tPηt = ⟨ ∇ ∗ φ − φ − φe − φe + ∇ φe − µe, ∇ ∂ ∗ φ⟩ A|| x[(Pηt ) (Pηt )] A|| xPηt h x tPηt for all x ∈ 5Q and t > 0 and then write ¨ = 2/ ⟨∇ ∗ φ − φ − φe − φe , ∗∇ ∂ ∗ φ⟩Ψ2 III (u1 J) x[(Pηt ) (Pηt )] A|| x tPηt δ dxdt n+1 ¨R+ + 2/ ⟨ ∇ φe − µe, ∇ ∂ ∗ φ⟩Ψ2 = + . (u1 J) A|| xPηt h x tPηt δ dxdt : III1 III2 n+1 R+ To estimate III , we integrate by parts in x with respect to the measure µ to write ¨ 1 = 2/ ∗ φ − φ − φe − φe ∗ ∂ ∗ φ Ψ2 µ III1 (u1 J)[(Pηt ) (Pηt )](L||,µ tPηt ) δ d dt n+1 ¨R+ − ∗ φ − φ − φe − φe ⟨∇ 2Ψ2/ , ∗∇ ∂ ∗ φ⟩ [(Pηt ) (Pηt )] x(u1 δ J) A|| x tPηt dxdt n+1 R+ = ′ + ′′, : III1 III1 ∗ which is justified by the definition of L||,µ (recall (2.8), (2.9) and (4.16)). 32 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

′ To estimate III1, we use Hardy’s inequality (see, for instance, page 272 in [S1]) 2 ∗ 2 −t L||,µ −t L||,µ to observe, for the semigroups Pt ∈ {e , e }, the estimate ˆ ˆ (ˆ ) ˆ ∞ ∞ ηt 2 ∞ 2 dt dt 2 dt 2 n |Pη − | ≤ |∂ P | ≲ |∂ P | ∀ ∈ R . t f f 3 s s f ds 3 t t f f Lµ( ) 0 t 0 0 t 0 t ∥1 ∥ ≲ ∥ ∥ ≂ Ψ ⊆ Ω We then recall that Ωη,Q u1 ∞ u ∞ and J 1 on supp δ η/8,Q,δ to obtain ˆ ˆ ∞ | ′ | ≲ ∥ ∥2 | ∗ φ − φ| + | φe − φe| | ∗ ∂ ∗ φ| µ III1 u ∞ ( Pηt Pηt ) L||,µ tPηt dtd Rn ( 0 ) ( ) ˆ ˆ ∞ 1/2 ˆ ∞ 1/2 2 ∗ 2 2 dt 2 ∗ ∗ 2 dt ≲ ∥ ∥ | φ − φ| + | η φe − φe| | ∂ φ| µ u ∞ Pηt P t 3 t L||,µ tPηt d Rn t t ( 0 ) ( 0 ) ¨ 1/2 ¨ 1/2 ≲ ∥ ∥2 |∂ ∗φ|2 + |∂ φe|2 µdt | 2∂ ∗ ∗ φ|2 µdt u ∞ tPt tPt d t tL||,µPηt d n+1 n+1 R+ t R+ t 2 2 2 1/2 2 ≲ ∥u∥ (∥∇φ∥ n n + ∥∇φe∥ n n ) ∥∇φ∥ 2 Rn,Rn ≲ ∥u∥ µ(Q), ∞ Lµ2(R ,R ) Lµ2(R ,R ) Lµ( ) ∞ where the ﬁnal line uses the vertical square function estimates from (2.13)-(2.14) L ∈ { ∗ } 1,2 φ φe for µ L||,µ, L||,µ and the W0,µ(5Q)-Hodge estimates for , from (4.14)-(4.15). ′′ | ∗ φ−φ| ≲ | φe−φe| ≲ Ψ ⊆ Ω To estimate III1 , recall that Pηt t and Pηt t on supp δ η/8,Q,δ ≂ |∇ −1 | ≲ |∇ ∂ ∗ φ| ∇ 2 Ψ2 by (4.19), whilst J 1 and x(J ) x tPηt , so distributing x over u1, δ / ′ and 1 J yields terms that can be controlled in the same way J, K and S12 in (4.37). To estimate III2, note that the estimates used to control φ and Pηtφ also hold for e φe and Pηtφe by (4.14)-(4.15) and (4.18)-(4.19), whilst divµ h = divµ h = 0 by (4.13), hence III2 can be estimated in the˜ same way as II in (4.44). ′′ 2 −1 2 |III | + |III | ≲ σ n+1 ⟨A ∇u , ∇u ⟩Ψ tdxdt + σ ∥u∥ µ Q This gives 1 2 R+ 1 1 1 δ ∞ ( ), hence ¨ 2 −1 2 (4.46) S2 ≲ σ ⟨A1∇u1, ∇u1⟩Ψδ tdxdt + σ ∥u∥∞µ(Q) ∀σ ∈ (0, 1). n+1 R+

We combine (4.36), (4.39) and (4.46) to obtain (4.33), as required. □

5. Solvability of the Dirichlet Problem

This section is dedicated to the proof of Theorem 1.2. We ﬁrst consider the construction and properties of a degenerate elliptic measure ωX for degenerate el- n+1 liptic equations div(A∇u) = 0 in the upper half-space, where X = (x, t) ∈ R+ and n ≥ 2. The t-independent coeﬃcient matrix A is assumed throughout to satisfy the degenerate bound and ellipticity in (1.1) for some constants 0 < λ ≤ Λ < ∞ n and an A2-weight µ on R . This is necessary as the literature only seems to treat bounded domains whilst the passage to unbounded domains in the uniformly elliptic case (see Section 10 in [LSW] and [HK]) relies on a global version of the Sobolev embedding in (2.4), which is not known for A2-weights in general. The degenerate elliptic measure is then shown to be in the A∞-class with respect to µ on the boundary Rn in Theorem 5.30 and the solvability of the Dirichlet problem follows in Theorem 5.34. These results together prove Theorem 1.2. CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 33

5.1. Boundary estimates for solutions. We require some estimates for solutions near the boundary ∂Σ of a bounded Lipschitz domain Σ ⊂ Rn (see Section 2 of [CFMS] for the standard definition). These estimates require some regularity on the domain boundary but no attempt is made here to obtain the minimal such regularity, as the focus is to define and analyse a degenerate elliptic measure on Rn. The Lipschitz regularity of the boundary ∂Σ ensures that the smooth class C∞(Σ) 0,1 1,2 and the Lipschitz class C (Σ) are both dense in Wµ (Σ) (see Theorem 3.4.1 in [Mo] and page 29 in [KS]). This allows the usual definition, for E ⊆ ∂Σ 1,2 1,2 and u ∈ Wµ (Σ), whereby u ≥ 0 on E in the Wµ (Σ)-sense means there exists 0,1 1,2 a sequence u j in C (Σ) that converges to u in Wµ (Σ) with u j(x) ≥ 0 for all x ∈ E. This induces definitions for inequalities ≤, ≥ and =, between functions 1,2 and/or constants, on E in the Wµ (Σ)-sense (see, for instance, Definition 5.1 in 1,2 [KS]). Moreover, with sup∂Σ u := inf{k ∈ R : u ≤ k on ∂Σ in the Wµ (Σ)-sense} and inf∂Σ := − sup∂Σ(−u), the weak maximum principle holds (see Theorem 2.2.2 in [FKS]), and the strong version follows by the Harnack inequality in (2.18) (see Corollary 2.3.10 in [FKS]). We can now state a Holder¨ continuity estimate and a Harnack inequality for certain solutions near the boundary. For a cube Q ⊂ Rn, recall the corkscrew point n+1 XQ := (xQ, ℓ(Q)) and denote the Carleson box in R+ by TQ := Q×(0, ℓ(Q)). Also, n+1 1,2 recall that µ(x, t):= µ(x), so dµ(x, t) = µ(x)dxdt, for (x, t) ∈ R . If u ∈ Wµ (T2Q) 1,2 is a solution of div(A∇u) = 0 in T2Q, and u = 0 on 2Q in the Wµ (T2Q)-sense, then ( ) ( )α 1/2 | , | ≲ t | |2 µ ∀ , ∈ , (5.1) u(x t) ℓ u d (x t) TQ (Q) T2Q and if, in addition, u ≥ 0 almost everywhere on T2Q, then

(5.2) u(X) ≲ u(XQ) ∀X ∈ TQ, α λ Λ µ where is from (2.17) and the implicit constants depend only on n, , and [ ]A2 . Estimate (5.1) follows from standard reﬂection arguments and the interior Holder¨ continuity estimate in (2.17), as observed on page 102 in [FKS]. Estimate (5.2) can then be deduced from (5.1) and the interior Harnack inequality in (2.18), as in the uniformly elliptic case (see the proof of Theorem 1.1 in [CFMS], which does not use the assumption therein that A is symmetric).

5.2. Definition and properties of degenerate elliptic measure. For X ∈ Rn+1, x ∈ Rn and r > 0, we use B(X, r):= {Y ∈ Rn+1 : |Y − X| < r} to denote balls in Rn+1 and ∆(x, r):= {y ∈ Rn : |x| < r} to denote balls in Rn, where ∆(x, r) is identified n+1 n+1 with the surface ball B((x, 0), r) ∩ ∂R+ in R . For each R > 0, consider the n+1 bounded Lipschitz domain ΣR := B(0, R) ∩ R+ with Lipschitz constant at most 1. ∈ Σ ωX ∂Σ For each X R, the degenerate elliptic measure´ R is the measure on R, as defined on page 583 in [FJK2], such that u(X) = h dωX solves the Dirichlet ∂ΣR R problem for continuous boundary data h ∈ C(∂ΣR) in the sense that div(A∇u) = 0 Σ ∈ Σ | = in R and u C( R) with u ∂ΣR h. n n We now define the degenerate elliptic measure on R . If f ∈ Cc(R ), fix R0 > 0 n+1 such that supp f ⊆ ∆(0, R0) and set f equal to zero on R+ , so then f ∈ C(∂ΣR) 34 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS for all R ≥ R , where f (X):= max{ f (X), 0}, thus 0 ˆ = ωX ∀ ∈ Σ uR(X): f d R X R ∂ΣR solve the Dirichlet problem as above in ΣR for all R ≥ R0. The maximum principle ≤ ≤ ≤ ∈ Σ then implies that uR (X) uR (X), whenever R0 R1 R2 and X R1 , and that ∥ ∥ ≤ ∥ ∥1 2 supR>0 uR ∞ f ∞. This allows us to define = + − − ∀ ∈ Rn+1 (5.3) u(X): lim [uR(X) uR(X)] X + R→∞ n and since the mapping f 7→ u(X) is a positive linear functional on Cc(R ), the Riesz Representation Theorem implies that there exists a regular Borel´ probability ωX Rn = ωX measure (the degenerate elliptic measure) on such that u(X) Rn f d . n+1 The function u from (5.3) solves div(A∇u) = 0 in R+ . To prove this, note n+1 that ∥u∥∞ ≤ ∥ f ∥∞, so for each compact set K ⊂ R+ , the Holder¨ continuity of solutions in (2.17) ensures the equicontinuity required to apply the Arzela–Ascoli`

Theorem and extract a subsequence uR j that converges to u uniformly on K. This 1,2 combined with Caccioppoli’s inequality shows that uR j converges to u in Wµ (K), ∈ 1,2 Rn+1 φ ∈ ∞ Rn+1 = φ ⊂ Σ hence u Wµ,loc( + ). Moreover, if Cc ( + ) and K supp R, then ˆ

1/2 ⟨A∇ u − u , ∇φ⟩ ≤ Λ∥∇φ∥∞µ K ∥u − u ∥ 1,2 , (5.4) ( R) ( ) R Wµ (K) K ´ n+1 ⟨A∇u, ∇φ⟩ = from which it follows that R+ 0, as required. We note by (5.3) that, when restricted to any bounded Borel subset of Rn, the ωX ωX measures R converge weakly to , so Theorem 1 on page 54 of [EG] shows that (5.5) ωX(U) ≤ lim inf ωX(U), ωX(K) ≥ lim sup ωX(K), ωX(B) = lim ωX(B) →∞ R R →∞ R R R→∞ R for all bounded open sets U ⊂ Rn, all compact sets K ⊂ Rn, and all bounded Borel sets B ⊂ Rn such that ωX(∂B) = 0. This construction of the degenerate elliptic measure also provides for the following expected properties. n+1 n X Lemma 5.6. If X0, X1 ∈ R+ and E ⊆ R is a Borel set, then ω 0 (E) = 0 if and only if ωX1 (E) = 0. Moreover, the non-negative function u(X):= ωX(E) is a n+1 solution of div(A∇u) = 0 in R+ and the boundary Höldercontinuity estimate ( )α t (5.7) |u(x, t)| ≲ u(X ) ∀(x, t) ∈ T ℓ(Q) Q Q holds on all cubes Q such that 2Q ⊆ Rn \ E, where α is from (2.17) and the implicit λ Λ µ constants depend only on n, , and [ ]A2 , Proof. The proof follows that of Lemma 1.2.7 in [K], except we must account for n+1 the fact that the solution to the Dirichlet problem in R+ defined by (5.3) requires boundary data to have compact support, which is easily done as we now show. Suppose that ωX0 (E) = 0 and that K ⊆ E is a compact set. The regularity of the measure implies that ωX0 (K) = 0 and, for each ϵ > 0, there exists a bounded open set U ⊃ K such that ωX0 (U) < ϵ. In particular, we may assume that U is n bounded because K is compact, so by Urysohn’s Lemma there exists g ∈ Cc(R ) such that g(x) = 1 on K, 0 ≤ g(x) ≤ 1 on U, and supp g ⊂ U. It follows that CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 35 ´ = ωX Rn+1 u(X) Rn g d is the solution to the Dirichlet problem in + defined by (5.3) with boundary data g. Applying the Harnack inequality from (2.18) and connecting X0 with X1 via a Harnack chain then shows that there exists C > 0, depending on X0 and X1, such that

X1 X0 ω (K) ≤ u(X1) ≤ Cu(X0) ≤ Cω (U) ≤ Cϵ ∀ϵ > 0, hence ωX1 (K) = 0 for all compact sets K ⊆ E, and so ωX1 (E) = 0 by regularity. X n+1 The proof that u(X):= ω (E) is a solution of div(A∇u) = 0 in R+ also follows that of Lemma 1.2.7 in [K]. It remains to prove that the boundary Holder¨ continuity estimate holds on all cubes Q such that 2Q ⊆ Rn \ E. We ﬁrst consider when E is bounded. In that case, let Uδ denote the open δ-neighbourhood of E and set χ = φ ∗ 1 δ > ϵ > φ = ϵ−nφ /ϵ φ ∈ ∞ ∆ , ϵ,δ : ϵ Uδ for all 0, where´ ϵ(x): (x ) and Cc ( (0 1)) φ = is a ﬁxed non-negative function with Rn 1. In particular, since Uδ is open, we + have 1 ≤ 1 ≤ lim infϵ→ χϵ,δ. Consequently, if X = (x, t) ∈ Rn 1, then E Uδ 0 ˆ +ˆ X X X X (5.8) u(X) = ω (E) ≤ ω (Uδ) ≤ lim inf χϵ,δ dω ≤ lim inf χϵ,δ dω . Rn ϵ→0 ϵ→0 Rn χ ∞ Rn ∞ Rn+1 The function ϵ,δ belongs to Cc ( ) and thus extends to a function in Cc ( ). The construction of the degenerate elliptic measure (see pages 580–583 in [FJK2], which was the starting´ point for our extension to the upper half-space above) thus = χ ωX 1,2 3 implies that vϵ(X): Rn ϵ,δ d is in W (T 3 ) and vanishes on Q whenever 2 Q 2 0 < ϵ < δ < ℓ(Q)/4, so estimate (5.8) combined with the boundary Holder¨ continuity estimate in (5.1) and the boundary Harnack inequality in (5.2) shows that ( )α t (5.9) u(x, t) ≤ lim inf vϵ(x, t) ≲ lim inf vϵ(XQ) ∀(x, t) ∈ TQ. ϵ→0 ℓ(Q) ϵ→0 ϵ χ ≤ 1 We now let Uδ,ϵ denote the open -neighbourhood of Uδ, in which case ϵ,δ Uδ,ϵ and vϵ(X) ≤ ωX(Uδ,ϵ), so by (5.9) and the regularity of the degenerate elliptic measure we have ( )α ( )α t t XQ XQ u(x, t) ≲ lim inf ω (Uδ,ϵ) ≲ ω (Uδ) ∀(x, t) ∈ TQ. ℓ(Q) ϵ→0 ℓ(Q) This proves (5.7) if E is bounded, since the regularity of the measure also implies X X that ω Q (Uδ) approaches ω Q (E) = u(XQ) as δ approaches 0. If E is not bounded, = 1 ∈ N then applying (5.7) on the bounded sets Ek : 2k+1Q\2k QE, for k , shows that ∞ ∞ ( )α ( )α ∑ ∑ t t u(x, t) = ωX(E ) ≲ ωXQ (E ) = ωXQ (E) ∀(x, t) ∈ T , k ℓ(Q) k ℓ(Q) Q k=1 k=1 as required. □

5.3. Preliminary estimates for degenerate elliptic measure. In the uniformly elliptic case, there is a rich theory for the Green’s function on bounded domains, and speciﬁcally, estimates and connections with elliptic measure (see, for instance, Theorem 1.2.8 and Corollary 1.3.6 in [K]). This theory also extends to unbounded domains (see Section 10 in [LSW] and [HK]). In the degenerate elliptic case, the theory was developed on bounded domains in [FJK1], [FJK2] and [FKS], but it is not clear if there is always such a Green’s function on unbounded domains. In particular, the construction in [HK] for the uniformly elliptic case relies on the 36 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

(unweighted) global version of the Sobolev embedding in (2.4), which is not known for a general A2-weight. In what follows, we combine the properties of the Green’s n+1 function on the bounded domain ΣR := B(0, R) ∩ R+ with the limit properties in (5.5) to deduce estimates for degenerate elliptic measure on Rn. These will be used to prove Lemma 5.24 and ultimately Theorem 5.30.

For each R > 0, the Green’s function gR : ΣR × ΣR 7→ [0, ∞] is constructed by following Proposition 2.4 in [FJK1]. In particular, for each Y ∈ ΣR, the mapping X 7→ gR(´X, Y) is the Holder¨ continuous function in ΣR\{Y} that vanishes on ∂ΣR and satisﬁes ⟨A∇g (·, Y), ∇Φ⟩ = Φ(Y) for all Φ ∈ C∞(Σ ). As explained on page ΣR R c R 583 in [FJK2], these properties are valid on any NTA domain, hence a fortiori on ΣR. The proofs do not rely on the assumption therein that A is symmetric, although the symmetry property “gR(X, Y) = gR(Y, X)” is no longer guaranteed, as ∗ , = , − ∗∇ gR(X Y): gR(Y X) is the Green’s function for the adjoint operator div(A ). We will rely on the following two lemmas, which are immediate from Theorem 4 and Lemma 3 in [FJK2], respectively, to estimate the Green’s function gR and the degenerate elliptic measure ωR on ΣR.

Lemma 5.10. If X, Y ∈ ΣR and |X − Y| < dist(Y, ∂ΣR)/2, then ˆ ,∂Σ dist(Y R) s2 ds gR(X, Y) ≂ , |X−Y| µ(B(Y, s)) s λ Λ µ where the implicit constants depend only on n, , and [ ]A2 . n Lemma 5.11. If R > 0 and Q is a cube in R such that T2Q ⊂ ΣR, then , ℓ ωY gR(XQ Y) ≂ ωY (Q) = R(Q) ∀ ∈ Σ \ , R(Q) Y R T2Q ℓ(Q) µ(TQ) µ(Q) λ Λ µ where the implicit constants depend only on n, , and [ ]A2 . ωX ωX ≤ The degenerate elliptic measure R satisﬁes the doubling property R (2Q) ωX Rn ⊂ Σ ∈ Σ \ C0 R (Q) for all cubes Q in such that T2Q R and all X R T2Q, where > λ Λ µ the doubling constant C0 0 depends only on n, , and [ ]A2 . This is proved in Lemma 1 on page 584 of [FJK2] by using the estimates in Lemma 5.11, the Har- nack inequality in (2.18), and the doubling property of µ. The doubling constant C0 does not depend on R, which allows us to use the inequalities in (5.5) to show that the degenerate elliptic measure ωX is locally doubling on Rn, in the sense that (5.12) ωX(2Q) ≤ lim inf ωX(2Q) ≲ lim inf ωX( 1 Q) ≤ lim sup ωX( 1 Q) ≤ ωX(Q) →∞ R →∞ R 2 R 2 R R R→∞ ⊂ Rn ∈ Rn+1 \ 2 for all cubes Q and all X + T2Q, where the implicit constant is C0. In particular, the doubling property implies that ωX(∂Q) = 0 for all cubes Q ⊂ Rn (see page 403 in [GR] or Proposition 6.3 in [HM]), so (5.12) actually improves to X X ω (2Q) ≤ C0ω (Q), since by the equality in (5.5) we now have ωX = ωX (5.13) (Q) lim R (Q) R→∞ n n+1 for all cubes Q ⊂ R and all X ∈ R+ \ T2Q. This provides the following estimate for degenerate elliptic measure. Lemma 5.14. If Q is a cube in Rn, then ωXQ (Q) ≳ 1, where the implicit constant λ Λ µ depends only on n, , and [ ]A2 . CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 37

Rn > ⊂ Σ Proof. Let Q denote a cube in and ﬁx R0 0 such that T2Q R0 . The Holder¨ continuity at the boundary in (5.1) and the Harnack inequality in (2.18) imply (see the proof of Lemma 3 on page 585 in [FJK2]) that ωXQ ≳ ∀ ≥ , R (Q) 1 R R0 λ Λ µ where the implicit constant depends only on n, , and [ ]A2 , and so does not depend on R. The result follows by using Harnack’s inequality to shift the pole X ωXQ = ω Q ≳ □ (from X2Q to XQ) in (5.12)-(5.13) to obtain (Q) limR→∞ R (Q) 1. The estimates in Lemma 5.11 also imply the following Comparison Principle. The result is stated on page 585 in [FJK2] and the proof is the same as in the uniformly elliptic case (see Theorem 1.4 in [CFMS] or Lemma 1.3.7 in [K], neither of which use the assumption therein that A is symmetric). Lemma 5.15. (Comparison Principle) Let Q denote a cube in Rn and suppose that 1,2 u, v ∈ Wµ (T2Q) ∩ C(T2Q) with u, v ≥ 0 on T2Q . If div(A∇u) = div(A∇v) = 0 in T2Q and u = v = 0 on 2Q, then

u(X) u(XQ) ≂ ∀X ∈ TQ, v(X) v(XQ) λ Λ µ where the implicit constants depend only on n, , and [ ]A2 . The following corollary of these preliminaries will be used in Lemma 5.18 to estimate Radon–Nikodym derivatives of the degenerate elliptic measure. n Lemma 5.16. If Q0 and Q are cubes in R such that Q ⊆ Q0, then X ω (Q) + ωXQ0 ≂ ∀ ∈ Rn 1 \ , (Q) X X + T2Q0 ω (Q0) λ Λ µ where the implicit constants depend only on n, , and [ ]A2 . ⊂ Rn ∈ Rn+1 \ Proof. Let Q Q0 be cubes in , suppose that X + T2Q0 and consider > ∈ Σ ⊂ Σ R 0 large enough so that X R and T4Q0 R. Lemma 5.11 shows that ωX ℓ ≂ µ , , R (Q0) (Q0) (Q0) gR(XQ0 X) ωX ℓ ≂ µ , R (Q) (Q) (Q) gR(XQ X) X ω 3Q0 ℓ ≂ µ , . R (Q) (Q) (Q) gR(XQ X3Q0 ) = , = , ∇ = ∇ = If u(Y) gR(Y X) and v(Y) gR(Y X3Q0 ), then div(A u) div(A u)v 0 in T2Q0 and u = v = 0 on 2Q0, so the Comparison Principle in Lemma 5.15 shows that g (X , X) u(X ) u(X ) g (X , X) R Q = Q ≂ Q0 = R Q0 . , , gR(XQ X3Q0 ) v(XQ) v(XQ0 ) gR(XQ0 X3Q0 ) , ≂ ℓ /µ Also, Lemma 5.10 shows that gR(XQ0 X3Q0 ) (Q0) (Q0), so together we obtain X ω (Q) g (X , X) µ(Q) ℓ(Q ) g (X , X ) µ(Q) ℓ(Q ) X R ≂ R Q 0 ≂ R Q 3Q0 0 ≂ ω 3Q0 (Q). ωX , ℓ µ , ℓ µ R R (Q0) gR(XQ0 X) (Q) (Q0) gR(XQ0 X3Q0 ) (Q) (Q0) X ωX ≂ ωX ω Q0 The Harnack inequality from (2.18) then shows that R (Q) R (Q0) R (Q) and the result follows by using (5.13) to estimate the limit as R approaches inﬁnity. □ 38 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

n+1 X X If X, X0 ∈ R+ , then Lemma 5.6 shows that ω and ω 0 are mutually absolutely continuous, so the Lebesgue differentiation theorem for the locally doubling measure ωX0 implies that the Radon–Nikodym derivative of ωX satisfies dωX ωX(Q(y, s)) X0 n (5.17) K(X0, X, y):= (y) = lim ω -a.e. y ∈ R , dωX0 s→0 ωX0 (Q(y, s)) where Q(y, s) denotes the cube in Rn with centre y and side length s. The following decay estimate for the kernel function K extends Lemma 2 on page 584 in [FJK2]. It is the final property of degenerate elliptic measure needed to prove Lemma 5.24. n Proposition 5.18. If Q0 and Q are cubes in R such that Q ⊆ Q0, then { }−α 1 |y − xQ| , , ≲ , ωXQ0 ∈ , K(XQ0 XQ y) X max 1 -a.e. y Q0 ω Q0 (Q) ℓ(Q) α> λ Λ µ where 0 from (2.17) and the implicit constant depend only on n, , and [ ]A2 .

n J−1 J Proof. Let Q ⊆ Q0 denote cubes in R and ﬁx J ∈ N such that 2 Q ⊆ Q0 ⊆ 2 Q. If y ∈ Q, then Lemma 5.16 and the Harnack inequality in (2.18) show that X X ω 2Q0 (Q(y, s)) ω Q0 (Q(y, s)) ωXQ , ≂ ≂ (Q(y s)) X X ω 2Q0 (Q) ω Q0 (Q) whenever 0 < s < dist(y, Rn \ Q). If y ∈ 2 jQ \ 2 j−1Q for some j ∈ {1,..., J}, then the boundary Holder¨ continuity estimate in (5.7) combined with Lemma 5.16 and the Harnack inequality in (2.18) show that ( )α ( )α ℓ ℓ ωXQ0 , (Q) X − (Q) (Q(y s)) ωXQ , ≲ ω 2 j 2Q , ≂ (Q(y s)) j−2 (Q(y s)) X 2 ℓ(Q) |y − xQ| ω Q0 (2 jQ) whenever 0 < s < dist(y, Rn \ (2 jQ \ 2 j−2Q)), where α > 0 from (2.17) and the λ Λ µ implicit constants depend only on n, , and [ ]A2 . The result follows by using these two estimates to bound the limit as s approaches zero in (5.17). □

5.4. The A∞-estimate for degenerate elliptic measure. We now combine the properties of degenerate elliptic measure with good ϵ0-coverings for sets, as introduced in [KKoPT] and defined below (see also [KKiPT]), to construct bounded solutions that satisfy the truncated square function estimate in Lemma 5.24. This result, combined with the Carleson measure estimate from Theorem 1.3, allows us to prove the A∞-estimate for the degenerate elliptic measure in Theorem 5.30. This avoids the need to apply the method of ϵ-approximability, as was done in [HKMP], and so simplifies the proof in the uniformly elliptic case. Let D(Rn) denote the standard collection {2k( j + [0, 1]n): k ∈ Z, j ∈ Zn} of all closed dyadic cubes S in Rn. For each S ∈ D(Rn) and η = 2−K, where K ∈ N, define D(S ):= {S ′ ∈ D(Rn): S ′ ⊆ S } and (5.19) Dη(S ):= {S ′ ∈ D(S ): ℓ(S ′) = 2−Kℓ(S )}, so Dη(S ) is precisely the set of all dyadic descendants of S at scale 2−Kℓ(S ). n Definition 5.20. Suppose that Q0 is a cube in R . If ϵ0 > 0, k ∈ N, Q ⊆ Q0 is a ⊆ ϵ { }k cube and E Q, then a good 0-cover of E of length k in Q is a collection Ol l=1 of nested open sets that satisfy E ⊆ Ok ⊆ Ok−1 ⊆ ... ⊆ O1 ⊆ Q and each of which CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 39

= ∪∞ l { l} ⊆ D Rn has a decomposition Ol i=1S i given by a collection S i i∈N ( ) of dyadic cubes with pairwise disjoint interiors such that − − ωX2Q0 ∩ l 1 ≤ ϵ ωX2Q0 l 1 ∀ ∈ N, ∀ ∈ { ,..., }. (5.21) (Ol S i ) 0 (S i ) i l 2 k Let us record a few important consequences of this definition that will be needed. It is proved on page 243 in [KKoPT] that for each i ∈ N and l ∈ {2,..., k}, there ∈ N l l−1 ℓ l ≤ 1 ℓ l−1 exists a unique j such that S i is a proper subset of S j , thus (S i) 2 (S j ). Also, for m ∈ {2,..., k}, iterating (5.21) as in Lemma 2.5 of [KKoPT] shows that − ωX2Q0 ∩ m ≤ ϵl mωX2Q0 m ∀ ∈ N, ∀ ∈ { ,..., }. (5.22) (Ol S i ) 0 (S i ) i l m k In the uniformly elliptic case, the following result is Lemma 2.3 from [KKiPT]. The proof extends to the degenerate elliptic case, since it only relies on the fact that X2Q the degenerate elliptic measure ω 0 is doubling when restricted to the cube Q0. n Lemma 5.23. Suppose that Q0 is a cube in R . If ϵ0 > 0, then there exists δ0 > 0, ϵ λ Λ µ depending only on 0, n, , and [ ]A2 , such that the following property holds: X2Q If Q ⊆ Q0 is a cube and E ⊆ Q0 such that ω 0 (E) ≤ δ0, then there exists a good X2Q ϵ0-cover of E of length k in Q for some natural number k ≂ log(ω 0 (E))/log ϵ0, λ Λ µ where the implicit constants depend only on n, , and [ ]A2 . We can now prove the following lemma by adapting the proof in [KKiPT] to the degenerate elliptic case. The original argument has also been somewhat modified. n Lemma 5.24. Suppose that Q0 is a cube in R . If M ≥ 1, then there exists δM > 0, λ Λ µ depending only on M, n, , and [ ]A2 , such that the following property holds: X2Q If Q ⊆ Q0 is a cube and E ⊆ Q and ω 0 (E) ≤ δM, then there is a Borel subset n X n+1 B of R such that the solution u(X):= ω (B) of div(A∇u) = 0 in R+ satisfies ˆ ˆ γℓ(Q) dµ(y) dt M ≤ |t∇u(y, t)|2 ∀x ∈ E, 0 ∆(x,γt) µ(∆(x, t)) t γ > λ Λ µ where 0 is a constant that depends only on n, , and [ ]A2 .

Proof. We introduce three constants ϵ0, δ, η ∈ (0, 1) that will be chosen with δ ≤ δ0, −K where δ0 is determined by ϵ0 as in Lemma 5.23, and η = 2 for some K ∈ N. X2Q Therefore, if E ⊆ Q ⊆ Q0 and ω 0 (E) ≤ δ, then there exists a good ϵ0-cover of X2Q E of length k in Q such that k ≂ log(ω 0 (E))/ log ϵ0. This cover is denoted by { }k = ∪∞ l l Ol l=1 with Ol i=1S i as in Deﬁnition 5.20, and for each such cube S i, a dyadic el Dη l l descendant S i in (S i) that contains the centre of S i is now ﬁxed and e = ∪∞ el, (5.25) Ol : i=1S i where we note that ℓ(Sel) = ηℓ(S l) in accordance with (5.19). i i ∑ n k We claim that there exists a Borel subset B of R such that 1B = = 1 e \ . ∑ m 2 O j−1 O j k To see this, suppose that = 1 e (x) , 0 and let l denote the smallest integer m 2 O j−1\O j 0 e ∈ , 1 e = ∈ − \ < l [2 k] such that O − \O (x) 1. It must hold that x Ol0 1 Ol0 , so then x Ol0 , l 1 l e which implies that x < O and x < O for all l ≥ l , hence 1 e (x) = 0 for all l l 0 Ol−1\Ol l > l0 and the claim follows. 40 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

X n+1 We now aim to choose ϵ0, η ∈ (0, 1) such that u(X):= ω (B) on R+ satisﬁes | − | ≳ ∀bl ∈ Dη l , ∀ ∈ N, ∀ ∈ { ,..., }, (5.26) u(Xη l ) u(Xηbl ) 1 S i (S i) i l 1 k S i S i , λ, Λ µ where the implicit constant depends only on the allowed constants n and [ ]A2 , l l l bl and if xi andx ˆi denote the centres of S i and S i, then the relevant corkscrew points l l l 2 l are precisely Xη l = (x , ηℓ(S )) and Xηbl = (x ˆ , η ℓ(S )). To this end, we proceed S i i i S i i i to obtain estimates for u(Xη l ) and u(Xηbl ). S i S i To estimate u(Xη l ), write S i ˆ ˆ XηS l XηS l = 1 ω i + 1 ω i = + . u(XηS l ) B d B d : I II i Rn\ l l S i S i

XηS l α ≤ ω i Rn \ l ≤ η The boundary Holder¨ continuity in (5.7) shows that I ( S i) C0 , where C0, α > 0 depend only on the allowed constants. To estimate II, write ∑l ˆ ∑k ˆ ˆ XηS l XηS l XηS l = 1 e ω i + 1 e ω i + 1 e ω i II − \ d − \ d \ + d l O j 1 O j l O j 1 O j l Ol Ol 1 j=2 S i j=l+2 S i S i

=: II1 + II2 + II3. = ∈ { ,..., } l ⊆ ⊆ First, observe that II1 0, since if m 2 l , then S i Ol O j and so e \ ∩ l = (O j−1 O j) S i Ø. To estimate II2, the kernel function representation in (5.17) and estimates in Proposition 5.18, the local doubling property of the degenerate elliptic measure in (5.12) and property (5.22) of the good ϵ0-covering, show that ∑k ˆ = , , ωX2Q0 II2 K(X2Q0 XηS l y) d (y) e \ ∩ l i j=l+2 (O j−1 O j) S i ∑k ( ) Cη X e l ≤ ω 2Q0 − \ ∩ X l (O j 1 O j) S i ω 2Q0 (S ) i j=l+2 ∑k Cη X l ≤ ω 2Q0 − ∩ X l (O j 1 S i) ω 2Q0 (S ) i j=l+2 ∑k Cη j−1−l X l ≤ ϵ ω 2Q0 ≤ ηϵ / − ϵ , X l 0 (S i) C 0 (1 0) ω 2Q0 (S ) i j=l+2 where the constant Cη > 0 depends only on η and the allowed constants. l ∩ e = el e To estimate II3, observe that S i Ol S i by the deﬁnition of Ol in (5.25), hence ˆ ˆ XηS l XηS l ′ ′′ = ω i − ω i = − . II3 d d : II3 II3 el el∩ S i S i Ol+1 ′′ The term II3 is estimated in the same way as II2 above to show that

′′ Cη Cη ≤ ωX2Q0 ∩ el ≤ ωX2Q0 ∩ l ≤ ϵ . II (Ol+1 S i) (Ol+1 S i) Cη 0 3 ωX2Q l ωX2Q l 0 (S i) 0 (S i) CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 41

′ l l l el We estimate II from above and below. First, note that Xη l = (x , ηℓ(S )), x ∈ S 3 S i i i i i XηS l XeS l ℓ el = ηℓ l ω i el ≂ ω i el and (S i) (S i), so (S i) (S i) by the Harnack inequality in (2.18), XeS l ω i el ≳ ∈ , whilst (S i) 1 by Lemma 5.14. Thus, there exists c0 (0 1) depending only ′ XηS l = ω i el ≥ ﬀ on the allowed constants such that II3 (S i) c0. Next, choose a di erent l , el Dη l l dyadic descendant S i S i in (S i) that contains the centre of S i. The preceding e X X X ηS l l ηS l l el ηS l el argument shows that ω i (S ) ≥ c , whilst ω i (S ∩ S ) ≤ ω i (∂S ) = 0, hence ei 0 ei i i X X X ′ ηS l el ηS l n el ηS l l c ≤ II = ω i (S ) = 1 − ω i (R \ S ) ≤ 1 − ω i (S ) ≤ 1 − c . 0 3 i i ei 0 The above estimates together show that if ϵ0 ∈ (0, 1/2), then α (5.27) c0 ≤ u(Xη l ) ≤ C0η + 3Cηϵ0 + 1 − c0. S i

To estimate u(XηSbl ), write i ˆ ˆ XηbS l XηbS l = 1 ω i + 1 ω i = b+ b u(XηSbl ) B d B d : I II i Rn\bl bl S i S i as well as ∑l ˆ ∑k ˆ ˆ X X X b ηbS l ηbS l ηbS l = 1 e ω i + 1 e ω i + 1 e ω i II − \ d − \ d \ + d bl O j 1 O j bl O j 1 O j bl Ol Ol 1 j=2 S i j=l+2 S i S i b b b =: II1 + II2 + II3.

XηbS l α b≤ ω i Rn \ bl ≤ η The arguments used to estimate I, II1 and II2 show that I ( S i) C0 , b b b II1 = 0 and II2 ≤ Cηϵ0/(1 − ϵ0). To estimate II3, observe that bl ∩ e \ = bl ∩ el \ , S i (Ol Ol+1) (S i S i) Ol+1 X ηbS l bl el b bl el where either ω i (S ∩ S ) = 0 and II = 0, or S = S and i ˆ i ˆ 3 i i ′ ′′ XηbS l XηbS l b = ω i − ω i = b − b . II3 d d : II3 II3 bl bl∩ S i S i Ol+1 The boundary Holder¨ continuity estimate in (5.7) shows that ′ XηbS l XηbS l α b = ω i bl = − ω i Rn \ bl ≥ − η II3 (S i) 1 ( S i) 1 C0 ′′ whilst repeating the arguments used to estimate II3 shows that ′′ Cη Cη b ≤ ωX2Q0 ∩ bl ≤ ωX2Q0 ∩ l ≤ ϵ . II3 (Ol+1 S i) X (Ol+1 S i) Cη 0 ωX2Q bl ω 2Q0 l 0 (S i) (S i) These estimates together show that if ϵ ∈ (0, 1/2), then either 0 ( ) α α (5.28) 0 ≤ u(Xηbl ) ≤ C0η + 3Cηϵ0 or u(Xηbl ) ≥ 1 − C0η + Cηϵ0 . S i S i The estimates (5.27) and (5.28) together imply that α |u(Xη l ) − u(Xηbl )| ≥ c0 − 2C0η − 4Cηϵ0. S i S i α We thus obtain (5.26) by ﬁrst choosing η ∈ (0, 1) so that 2C0η ≤ c0/4 and then choosing ϵ0 ∈ (0, 1/2) (depending on η) so that 4Cηϵ0 ≤ c0/4. These choices of η and ϵ0, which depend only on the allowed constants, are now ﬁxed. 42 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS

To complete the proof, suppose that M ≥ 1 and x ∈ E, and recall that δ ∈ (0, δ0) remains to be chosen, where δ0 is now fixed by our choice of ϵ0 as in Lemma 5.23. k { k} ∈ k First, fix a cube S in S i i∈N such that x S . The remarks after Definition 5.20 ∈ { ,..., − } l { l} then imply that for each l 1 k 1 , there exists a unique cube S in S i i∈N ∈ l l+1 ⊂ l ℓ l+1 ≤ 1 ℓ l ∈ { ,..., } such that x S and S S , thus (S ) 2 (S ). Next, for each l 1 k , fix a dyadic descendant Sbl in Dη(S l) such that x ∈ Sbl. Observe that, for some τ ∈ (0, 1) sufficiently close to 1 and depending only on η τ l , the corkscrew points XηS l and XηSbl both belong to the dilate Qη of the cube l = { , ∈ Rn+1 | − | < 1 + η2 ℓ l , η2 ℓ l < < + η2 ℓ l } Qη : (y t) + : y x ∞ ( 2 4 ) (S ) 2 (S ) t (1 ) (S ) ffl l η2 l l with ℓ(Q ) = (1 + )ℓ(S ). Therefore, if c := l u, then the Moser-type estimate η 2 Qη in (2.16), the Poincare´ inequality in (2.5) and the doubling property of µ show that | − |2 ≲ | − l|2 + | − l|2 u(XηS l ) u(XηSbl ) u(XηS l ) c u(XηSbl ) c ≲ ∥ − l∥2 u c ∞ τ l L ( Qη) l 2 ≲η |u − c | dµ l Qη l 2 2 (5.29) ≲ ℓ(Qη) |∇u| dµ Ql η ˆ ℓ(S l) ≲ ( ) |∇ |2 µ η2 u d l l µ ∆(x, (1 + )ℓ(S )) Qη ¨ 2 dµ(y) dt ≲ |t∇u(y, t)|2 . l µ ∆ , Qη ( (x t)) t

ℓ l+1 ≤ 1 ℓ l ℓ l′ ≤ l−l′ ℓ l ′ ≥ Iterating the bound (S ) 2 (S ) shows that (S ) 2 (S ) when l l. 1 k This implies that the collection {Qη,..., Qη} has the bounded intersection property ∈ { ,..., } + 1 + l′ whereby for each l 1 k , there are at most 3 2 log2( η2 1)) such cubes Qη l′ l satisfying Qη ∩ Qη , Ø. This allows us to sum estimate (5.29) over l ∈ {1,..., k} and then apply (5.26) to obtain ¨ ˆ ˆ µ γℓ(Q) µ 2 d (y) dt 2 d (y) dt k ≲η |t∇u(y, t)| ≲ |t∇u(y, t)| ∪k l µ(∆(x, t)) t ∆ ,γ µ(∆(x, t)) t l=1Qη 0 (x t) for some γ > 0 that depends only on η > 0 and thus only on the allowed constants. X2Q −1 To conclude, recall that k ≂ log(ω 0 (E) )/ log(1/ϵ0) ≥ log(1/δ)/ log(1/ϵ0), X2Q since ω 0 (E) ≤ δ < 1. Therefore, the result follows by choosing δ ∈ (0, δ0] such that M ≤ log(1/δ), since δM := δ depends only on M and the allowed constants. □

We now combine the above technical lemma with the Carleson measure estimate from Theorem 1.3 to prove the main A∞-estimate for degenerate elliptic measure. Rn ∈ Rn+1 \ ω = ωX⌊ Theorem 5.30. Suppose that Q0 is a cube in . If X + TQ0 and : Q0 denotes the degenerate elliptic measure restricted to Q0, then ω ∈ A∞(µ) and the following equivalent properties hold: CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 43

(1) For each ϵ ∈ (0, 1), there exists δ ∈ (0, 1), depending only on ϵ, n, λ, Λ µ ⊆ and [ ]A2 , such that the following property holds: If Q Q0 is a cube and E ⊆ Q such that ω(E) ≤ δω(Q), then µ(E) ≤ ϵµ(Q). (2) The measure ω is absolutely continuous with respect to µ and there exists q ∈ (1, ∞) such that the Radon–Nikodym derivative k := dω/dµ satisﬁes, on all surface balls ∆ ⊆ Q0, the reverse H¨olderestimate ( ) 1/q kq dµ ≲ k dµ, ∆ ∆ λ Λ µ where q and the implicit constant depend only on n, , and [ ]A2 . , θ > λ Λ µ (3) There exist C 0, depending only on n, , and [ ]A2 , such that ( )θ µ(E) ω(E) ≤ C ω(Q) µ(Q)

for all cubes Q ⊆ Q0 and all Borel sets E ⊆ Q.

Proof. It is well-known that (1)–(3) are equivalent (see Theorem 1.4.13 in [K]). ﬃ = Moreover, by Lemma 5.16, it su ces to prove (1) when X X2Q0 . In that case, by Lemma 5.24, the Carleson measure estimate in Theorem 1.3, Fubini’s Theorem and the doubling property of µ, it follows that for each M ≥ 1, there exists δM > 0, depending only on M and the allowed constants, such that the following property holds: If Q ⊆ Q0 is a cube and E ⊆ Q such that ω(E) ≤ δMω(Q), then there exists n+1 a solution u of the equation div(A∇u) = 0 in R+ with ∥u∥∞ ≤ 1 such that ˆ ˆ ˆ γℓ(Q) dµ(y) dt Mµ(E) ≤ |t∇u(y, t)|2 dµ(x) ∆ ,γ µ(∆(x, t)) t ˆE 0 ˆ (x t) γℓ˜ (Q) dt ≲ |t∇u(y, t)|2 dµ(y) ≲ µ(Q), 0 γ˜Q t where the implicit constants andγ ˜ > γ > 0 depend only on the allowed constants. Therefore, if ϵ ∈ (0, 1), we choose M(ϵ) ≥ 1 and thus δM(ϵ) ∈ (0, 1), depending only on ϵ and the allowed constants, such that µ(E) ≤ ϵµ(Q), as required. □

5.5. The square function and non-tangential maximal function estimates. The p Lµ(Rn)-norm equivalence between the square function S u and the non-tangential maximal function N∗u of solutions u in Theorem 1.5 is now a corollary of the main A∞-estimate for the degenerate elliptic measure in Theorem 5.30. This was proved by Dahlberg, Jerison and Kenig in Theorem 1 of [DJK], which actually provides the more general result in Theorem 5.31 below. In particular, the degenerate elliptic case is treated on page 106 of [DJK], noting that the normalisation u(X0) = 0 assumed therein is actually only required for the so-called N ≲ S -estimate. Theorem 5.31. Suppose that Φ : [0, ∞)→[0, ∞) is an unbounded, non-decreasing, continuous function with Φ(0) = 0 and Φ(2t) ≤ CΦ(t) for all t > 0 and some C > 0. n+1 If div(A∇u) = 0 in R+ , then ˆ ˆ Φ(S u) dµ ≲ Φ(N∗u) dµ, Rn Rn 44 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS and if, in addition, u(X ) = 0 for some X ∈ Rn+1, then 0 ˆ 0 ˆ + Φ(N∗u) dµ ≲ Φ(S u) dµ, Rn Rn Φ λ Λ µ where the implicit constants depend only on X0, , n, , and [ ]A2 .

The next result is also a consequence of the main A∞-estimate in Theorem 5.30. It will allow us to construct solutions to the Dirichlet problem (D)p,µ as integrals p of Lµ(Rn)-boundary data with respect to degenerate elliptic measure. 1 + 1 = ∈ , ∞ Lemma 5.32. Suppose that p q 1, where q (1 ) is the reverse H¨older n+1 exponent from Theorem 5.30. If X = (x, t) ∈ R+ , then the Radon–Nikodym q derivative k(X, ·):= dωX/dµ is in Lµ(Rn) and ˆ k((x, t), y)q dµ(y) ≲ µ(∆(x, t))1−q. Rn ´ p ∈ Rn = ωX ∥ ∥ p ≲ ∥ ∥ p . Moreover, if f Lµ( ) and u(X): Rn f (y) d , then N∗u Lµ(Rn) f Lµ(Rn) λ Λ µ The implicit constant in each estimate depends only on n, , and [ ]A2 .

n+1 Proof. Suppose that X = (x, t) ∈ R+ . The proof of Proposition 5.18 shows that k((x, 2 jt), y) k((x, t), y) ≲ 2− jα ∀y ∈ ∆(x, 2 jt) \ ∆(x, 2 j−1t), ∀ j ∈ N. ω(x,2 jt)(∆(x, 2 jt)) Applying the reverse Holder¨ estimate from Theorem 5.30 then shows that ˆ k((x, t), y)q dµ(y) Rn ˆ ∑∞ ˆ = k((x, t), y)q dµ(y) + k((x, t), y)q dµ(y) ∆ , ∆ , j \∆ , j−1 (x t) j=1 (x 2 t) (x 2 t) ∑∞ ≲ µ(∆(x, t))1−q + 2− jαqµ(∆(x, 2 jt))1−q ≲ µ(∆(x, t))1−q. j=1 To obtain the non-tangential maximal function estimate, it suﬃces to consider the case when f ≥ 0, since in general we may then decompose f = f + − f − into its + − n positive and negative parts f , f ≥ 0. To this end, suppose that x0 ∈ R and that n+1 X = (x, t) ∈ R+ in order to write ∑∞ ∑∞ = 1∆ , + 1 j+1 j = f f (x0 2t) f ∆(x0,2 t)\∆(x0,2 t) : f j j=1 j=0 and deﬁne ˆ ˆ X u j(X):= f j(y) dω (y) = f j(y) k(X, y) dµ(y). Rn Rn The self-improvement property of the reverse Holder¨ estimate from Theorem 5.30 (see Theorem 1.4.13 in [K]) implies that there exists an exponent r > q such that ( ) / 1 r 1 (5.33) k((x, t), y)r dµ(y) ≲ k((x, t), y) dµ(y) ≤ ∆ ∆ µ(∆) for all surface balls ∆ ⊆ ∆(x, t/2). CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 45

Now suppose that X = (x, t) ∈ Γ(x0). To estimate u0, we apply the interior Harnack inequality in (2.18) followed by Holder’s¨ inequality and (5.33) to obtain ˆ

u0(x, t) ≂ u0(x, 6t) ≤ f (y) k((x, 6t), y) dµ(y) ∆(x0,2t) ( ) ( ) ′ ˆ 1/r ˆ 1/r ′ ≤ |k((x, 6t), y)|r dµ(y) f (y)r dµ(y) ∆(x0,2t) ∆(x0,2t) ( ) ′ ˆ 1/r −1/r′ r′ ≲ µ(∆(x0, 2t)) f (y) dµ(y) ∆(x0,2t) r′ 1/r′ ≤ [Mµ( f )(x0)] .

To estimate u j when j ∈ N, we apply the boundary Holder¨ continuity estimate from (5.7) and then proceed as in the estimate above to obtain ( )α t j − jα j+2 u j(x, t) ≲ u j(x0, 2 t) ≂ 2 u j(x0, 2 t) 2 jtˆ − jα j+2 ≤ 2 f (y) k((x0, 2 t), y) dµ(y) j+1 ∆(x0,2 t) ( ) ( ) ′ ˆ 1/r ˆ 1/r − jα j+2 r r′ ≤ 2 k((x0, 2 t), y) dµ(y) f (y) dµ(y) j+1 j+1 ∆(x0,2 t) ∆(x0,2 t) ( ) ′ 1/r ′ ≲ 2− jα f (y)r dµ(y) j+1 ∆(x0,2 t) − jα r′ 1/r′ ≤ 2 [Mµ( f )(x0)] .

r′ 1/r′ n The above estimates together show that N∗u(x0) ≲ [Mµ( f )(x0)] for all x0 ∈ R , ′ ′ < = ∥ ∥ p ≲ ∥ ∥ p □ and since r q p, it follows that N∗u Lµ f Lµ , as required.

We conclude the paper by using the preceding lemma to obtain solvability of the Dirichlet problem (D)p,µ. A uniqueness result is also obtained but only for solutions that converge uniformly to 0 at infinity. This restriction does not appear in the uniformly elliptic case (see Theorem 1.7.7 in [K]). It arises here because of the absence of a Green’s function for degenerate elliptic equations on unbounded domains (see Section 5.3) and it is not clear to us whether this can be improved. 1 + 1 = ∈ , ∞ Theorem 5.34. Suppose that p q 1, where q (1 ) is the reverse Hölder p exponent from Theorem 5.30. The Dirichlet problem for Lµ(Rn)-boundary data is p solvable in the sense that for each f ∈ Lµ(Rn), there exists a solution u such that  n+1 div(A∇u) = 0 in R+ , p n (D)p,µ N∗u ∈ Lµ(R ),  limt→0 u(·, t) = f, p where the limit converges in Lµ(Rn)-norm and in the non-tangential sense whereby n limΓ(x)∋(y,t)→(x,0) u(y, t) = f (x) for almost every x ∈ R . Moreover, if f has compact support, then there is a unique solution u of (D)p,µ that converges uniformly to 0 at →∞ ∥ ∥ ∞ n+1 = infinity in the sense that limR u L (R+ \B(0,R)) 0. 46 STEVE HOFMANN, PHI LE, ANDREW J. MORRIS ´ ∈ p Rn = ωX ∈ Rn+1 Proof. Suppose that f Lµ( ) and define u(X): Rn f d for all X + . n+1 n We first prove that div(A∇u) = 0 in R+ . Let ( f j) j denote a sequence´ in Cc(R ) p Rn = ωX that converges to f in Lµ( ) and consider the solutions u j(X): Rn f j d . The Lq(Rn)-estimate for the Radon–Nikodym derivative dωX/dµ from Lemma 5.32 and µ ∥ − ∥ ∞ ≲ ∥ − ∥ p ∈ N the doubling property of show that u j u L (K) µ,K f j f Lµ(Rn) for all j ⊂ Rn+1 2 Rn and any compact set K + , so u j converges to u in Lµ,loc( ). Moreover, Cacioppoli’s inequality and the arguments preceding (5.4) show that u j converges 1,2 Rn = Rn+1 to a solution v in Wµ,loc( ), so then u v is a solution in + as required . ∥ ∥ p ≲ ∥ ∥ p The non-tangential maximal function estimate N∗u Lµ(Rn) f Lµ(Rn) is given by Lemma 5.32. To prove the non-tangential convergence to the boundary datum, n+1 first recall that u j ∈ C(R+ ) with u j|Rn := f j, so limΓ(x)∋(y,t)→(x,0) u j(y, t) = f j(x) (see Section 5.2). We combine this fact with the bound

lim sup |u(y, t) − f (x)| ≤ |N∗(u − u j)(x)| + |( f − f j)(x)| Γ(x)∋(y,t)→(x,0) for all x ∈ Rn. For any η > 0, we then apply Chebyshev’s inequality and the non-tangential maximal function estimate from Lemma 5.32, to show that ({ }) µ x ∈ Rn : lim sup |u(y, t) − f (x)| > η Γ(x)∋(y,t)→(x,0) n n ≤ µ({x ∈ R : N∗(u − u )(x) > η/2}) + µ({x ∈ R : |( f − f )(x)| > η/2}) ( j ) j −p p p ≲ η ∥N∗(u − u )∥ p + ∥ f − f ∥ p j Lµ(Rn) j Lµ(Rn) −p p ≲ η ∥ f − f ∥ p . j Lµ(Rn) p n It follows, since f j converges to f in Lµ(R ), that limΓ(x)∋(y,t)→(x,0) u(y, t) = f (x) for ∈ Rn ∥ ·, − ∥ p almost every x , as required. The norm convergence limt→0 u( t) f Lµ(Rn) then follows by Lebesgue’s dominated convergence theorem.

It remains to prove that u is the unique solution satisfying lim|X|→∞ ∥u(X)∥∞ = 0 when f has compact support. In that case, ﬁx R0 > 0 such that f is supported in the n+1 surface ball ∆(0, R0). If X ∈ R+ and |X| > 2R0, then the reverse Holder¨ estimate in Theorem 5.30 showsˆ that |u(X)| ≤ | f (y)| k(X, y) dµ(y) ∆ , (0 R0) ( ) ˆ 1/q ≤ ∥ ∥ p , q µ f Lµ(Rn) k(X y) d (y) ∆(0,|X|/2) / ≲ ∥ ∥ p µ ∆ , | |/ 1 q , µ f Lµ(Rn) ( (0 X 2)) k(X y) d (y) ∆(0,|X|/2) − / ≤ ∥ ∥ p µ ∆ , | |/ 1 p, f Lµ(Rn) ( (0 X 2)) whilst limR→∞ µ(∆(0, R)) = ∞, since µ is in the A∞-class with respect to Lebesgue n R →∞ ∥ ∥ ∞ n+1 = measure on , thus limR u L (R+ \B(0,R)) 0. The maximum principle allows us to conclude that any solution of (D)p,µ with this decay must be unique. □ CARLESON MEASURE ESTIMATES AND THE DIRICHLET PROBLEM 47

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Steve Hofmann,Department of Mathematics,University of Missouri,Columbia, MO 65211, USA. E-mail address: [email protected]

Phi L. Le,Mathematics Department,Syracuse University,Syracuse, NY 13244, USA. E-mail address: [email protected]

Andrew J. Morris,School of Mathematics,University of Birmingham,Birmingham, B15 2TT, UK. E-mail address: [email protected]